The value of the determinant has many implications for the matrix. Thus, let A be a KK dimension matrix, the cofactor expansion along the i-th row is defined with the following formula: Similarly, the mathematical formula for the cofactor expansion along the j-th column is as follows: Where Aij is the entry in the i-th row and j-th column, and Cij is the i,j cofactor.if(typeof ez_ad_units!='undefined'){ez_ad_units.push([[300,250],'algebrapracticeproblems_com-banner-1','ezslot_2',107,'0','0'])};__ez_fad_position('div-gpt-ad-algebrapracticeproblems_com-banner-1-0'); Lets see and example of how to solve the determinant of a 33 matrix using cofactor expansion: First of all, we must choose a column or a row of the determinant. Step 1: R 1 + R 3 R 3: Based on iii. a bug ? This proves that cofactor expansion along the \(i\)th column computes the determinant of \(A\). Or, one can perform row and column operations to clear some entries of a matrix before expanding cofactors, as in the previous example. Determinant; Multiplication; Addition / subtraction; Division; Inverse; Transpose; Cofactor/adjugate ; Rank; Power; Solving linear systems; Gaussian Elimination; Calculate cofactor matrix step by step. Doing a row replacement on \((\,A\mid b\,)\) does the same row replacement on \(A\) and on \(A_i\text{:}\). Also compute the determinant by a cofactor expansion down the second column. The cofactor expansion theorem, also called Laplace expansion, states that any determinant can be computed by adding the products of the elements of a column or row by their respective cofactors. have the same number of rows as columns). See how to find the determinant of a 44 matrix using cofactor expansion. \end{split} \nonumber \]. Consider the function \(d\) defined by cofactor expansion along the first row: If we assume that the determinant exists for \((n-1)\times(n-1)\) matrices, then there is no question that the function \(d\) exists, since we gave a formula for it. . \nonumber \]. Multiply each element in any row or column of the matrix by its cofactor. Therefore, the \(j\)th column of \(A^{-1}\) is, \[ x_j = \frac 1{\det(A)}\left(\begin{array}{c}C_{ji}\\C_{j2}\\ \vdots \\ C_{jn}\end{array}\right), \nonumber \], \[ A^{-1} = \left(\begin{array}{cccc}|&|&\quad&| \\ x_1&x_2&\cdots &x_n\\ |&|&\quad &|\end{array}\right)= \frac 1{\det(A)}\left(\begin{array}{ccccc}C_{11}&C_{21}&\cdots &C_{n-1,1}&C_{n1} \\ C_{12}&C_{22}&\cdots &C_{n-1,2}&C_{n2} \\ \vdots &\vdots &\ddots &\vdots &\vdots\\ C_{1,n-1}&C_{2,n-1}&\cdots &C_{n-1,n-1}&C{n,n-1} \\ C_{1n}&C_{2n}&\cdots &C_{n-1,n}&C_{nn}\end{array}\right). Online Cofactor and adjoint matrix calculator step by step using cofactor expansion of sub matrices. Check out 35 similar linear algebra calculators . Cofactor Matrix Calculator The method of expansion by cofactors Let A be any square matrix. In fact, the signs we obtain in this way form a nice alternating pattern, which makes the sign factor easy to remember: As you can see, the pattern begins with a "+" in the top left corner of the matrix and then alternates "-/+" throughout the first row. 2. det ( A T) = det ( A). Then the matrix that results after deletion will have two equal rows, since row 1 and row 2 were equal. Instead of showing that \(d\) satisfies the four defining properties of the determinant, Definition 4.1.1, in Section 4.1, we will prove that it satisfies the three alternative defining properties, Remark: Alternative defining properties, in Section 4.1, which were shown to be equivalent. We showed that if \(\det\colon\{n\times n\text{ matrices}\}\to\mathbb{R}\) is any function satisfying the four defining properties of the determinant, Definition 4.1.1 in Section 4.1, (or the three alternative defining properties, Remark: Alternative defining properties,), then it also satisfies all of the wonderful properties proved in that section. There are many methods used for computing the determinant. The minor of an anti-diagonal element is the other anti-diagonal element. . The proof of Theorem \(\PageIndex{2}\)uses an interesting trick called Cramers Rule, which gives a formula for the entries of the solution of an invertible matrix equation. First suppose that \(A\) is the identity matrix, so that \(x = b\). Form terms made of three parts: 1. the entries from the row or column. Advanced Math questions and answers. This formula is useful for theoretical purposes. No matter what you're writing, good writing is always about engaging your audience and communicating your message clearly. The result is exactly the (i, j)-cofactor of A! These terms are Now , since the first and second rows are equal. This shows that \(d(A)\) satisfies the first defining property in the rows of \(A\). Cofactor expansions are most useful when computing the determinant of a matrix that has a row or column with several zero entries. (1) Choose any row or column of A. You can also use more than one method for example: Use cofactors on a 4 * 4 matrix but Solve Now . Determinant of a 3 x 3 Matrix Formula. In fact, one always has \(A\cdot\text{adj}(A) = \text{adj}(A)\cdot A = \det(A)I_n,\) whether or not \(A\) is invertible. Your email address will not be published. 2 For each element of the chosen row or column, nd its The cofactor matrix of a given square matrix consists of first minors multiplied by sign factors:. Determinant of a Matrix Without Built in Functions. Since we know that we can compute determinants by expanding along the first column, we have, \[ \det(B) = \sum_{i=1}^n (-1)^{i+1} b_{i1}\det(B_{i1}) = \sum_{i=1}^n (-1)^{i+1} a_{ij}\det(A_{ij}). Looking for a way to get detailed step-by-step solutions to your math problems? Expand by cofactors using the row or column that appears to make the computations easiest. One way to think about math problems is to consider them as puzzles. To compute the determinant of a \(3\times 3\) matrix, first draw a larger matrix with the first two columns repeated on the right. If you need your order delivered immediately, we can accommodate your request. Find the determinant of \(A=\left(\begin{array}{ccc}1&3&5\\2&0&-1\\4&-3&1\end{array}\right)\). We want to show that \(d(A) = \det(A)\). If you don't know how, you can find instructions. Cofactor may also refer to: . find the cofactor First we compute the determinants of the matrices obtained by replacing the columns of \(A\) with \(b\text{:}\), \[\begin{array}{lll}A_1=\left(\begin{array}{cc}1&b\\2&d\end{array}\right)&\qquad&\det(A_1)=d-2b \\ A_2=\left(\begin{array}{cc}a&1\\c&2\end{array}\right)&\qquad&\det(A_2)=2a-c.\end{array}\nonumber\], \[ \frac{\det(A_1)}{\det(A)} = \frac{d-2b}{ad-bc} \qquad \frac{\det(A_2)}{\det(A)} = \frac{2a-c}{ad-bc}. This cofactor expansion calculator shows you how to find the determinant of a matrix using the method of cofactor expansion (a.k.a. Then, \[\label{eq:1}A^{-1}=\frac{1}{\det (A)}\left(\begin{array}{ccccc}C_{11}&C_{21}&\cdots&C_{n-1,1}&C_{n1} \\ C_{12}&C_{22}&\cdots &C_{n-1,2}&C_{n2} \\ \vdots&\vdots &\ddots&\vdots&\vdots \\ C_{1,n-1}&C_{2,n-1}&\cdots &C_{n-1,n-1}&C_{n,n-1} \\ C_{1n}&C_{2n}&\cdots &C_{n-1,n}&C_{nn}\end{array}\right).\], The matrix of cofactors is sometimes called the adjugate matrix of \(A\text{,}\) and is denoted \(\text{adj}(A)\text{:}\), \[\text{adj}(A)=\left(\begin{array}{ccccc}C_{11}&C_{21}&\cdots &C_{n-1,1}&C_{n1} \\ C_{12}&C_{22}&\cdots &C_{n-1,2}&C_{n2} \\ \vdots&\vdots&\ddots&\vdots&\vdots \\ C_{1,n-1}&C_{2,n-1}&\cdots &C_{n-1,n-1}&C_{n,n-1} \\ C_{1n}&C_{2n}&\cdots &C_{n-1,n}&C_{nn}\end{array}\right).\nonumber\]. The cofactor expansion formula (or Laplace's formula) for the j0 -th column is. Check out our solutions for all your homework help needs! 2 For. The definition of determinant directly implies that, \[ \det\left(\begin{array}{c}a\end{array}\right)=a. The remaining element is the minor you're looking for. The sign factor is equal to (-1)2+1 = -1, so the (2, 1)-cofactor of our matrix is equal to -b. Lastly, we delete the second row and the second column, which leads to the 1 1 matrix containing a. As an example, let's discuss how to find the cofactor of the 2 x 2 matrix: There are four coefficients, so we will repeat Steps 1, 2, and 3 from the previous section four times. It's a great way to engage them in the subject and help them learn while they're having fun. Let us review what we actually proved in Section4.1. \nonumber \], We computed the cofactors of a \(2\times 2\) matrix in Example \(\PageIndex{3}\); using \(C_{11}=d,\,C_{12}=-c,\,C_{21}=-b,\,C_{22}=a\text{,}\) we can rewrite the above formula as, \[ A^{-1} = \frac 1{\det(A)}\left(\begin{array}{cc}C_{11}&C_{21}\\C_{12}&C_{22}\end{array}\right). At every "level" of the recursion, there are n recursive calls to a determinant of a matrix that is smaller by 1: T (n) = n * T (n - 1) I left a bunch of things out there (which if anything means I'm underestimating the cost) to end up with a nicer formula: n * (n - 1) * (n - 2) . As shown by Cramer's rule, a nonhomogeneous system of linear equations has a unique solution iff the determinant of the system's matrix is nonzero (i.e., the matrix is nonsingular). \end{align*}. Indeed, when expanding cofactors on a matrix, one can compute the determinants of the cofactors in whatever way is most convenient. The cofactor matrix of a given square matrix consists of first minors multiplied by sign factors: More formally, let A be a square matrix of size n n. Consider i,j=1,,n. Putting all the individual cofactors into a matrix results in the cofactor matrix. For example, here we move the third column to the first, using two column swaps: Let \(B\) be the matrix obtained by moving the \(j\)th column of \(A\) to the first column in this way. 3 Multiply each element in the cosen row or column by its cofactor. That is, removing the first row and the second column: On the other hand, the formula to find a cofactor of a matrix is as follows: The i, j cofactor of the matrix is defined by: Where Mij is the i, j minor of the matrix. In this way, \(\eqref{eq:1}\) is useful in error analysis. Determinant by cofactor expansion calculator can be found online or in math books. First we expand cofactors along the fourth row: \[ \begin{split} \det(A) \amp= 0\det\left(\begin{array}{c}\cdots\end{array}\right)+ 0\det\left(\begin{array}{c}\cdots\end{array}\right) + 0\det\left(\begin{array}{c}\cdots\end{array}\right) \\ \amp\qquad+ (2-\lambda)\det\left(\begin{array}{ccc}-\lambda&2&7\\3&1-\lambda &2\\0&1&-\lambda\end{array}\right). I need help determining a mathematic problem. The calculator will find the matrix of cofactors of the given square matrix, with steps shown. Omni's cofactor matrix calculator is here to save your time and effort! The calculator will find the matrix of cofactors of the given square matrix, with steps shown. The Laplacian development theorem provides a method for calculating the determinant, in which the determinant is developed after a row or column. Cofactor Expansion Calculator Conclusion For each element, calculate the determinant of the values not on the row or column, to make the Matrix of Minors Apply a checkerboard of minuses to 824 Math Specialists 9.3/10 Star Rating $\endgroup$ One way to think about math problems is to consider them as puzzles. I hope this review is helpful if anyone read my post, thank you so much for this incredible app, would definitely recommend. This cofactor expansion calculator shows you how to find the determinant of a matrix using the method of cofactor expansion (a.k.a. The only hint I have have been given was to use for loops. Let \(A\) be the matrix with rows \(v_1,v_2,\ldots,v_{i-1},v+w,v_{i+1},\ldots,v_n\text{:}\) \[A=\left(\begin{array}{ccc}a_11&a_12&a_13 \\ b_1+c_1 &b_2+c_2&b_3+c_3 \\ a_31&a_32&a_33\end{array}\right).\nonumber\] Here we let \(b_i\) and \(c_i\) be the entries of \(v\) and \(w\text{,}\) respectively. Expert tutors will give you an answer in real-time. Cofactor expansion calculator - Cofactor expansion calculator can be a helpful tool for these students. For more complicated matrices, the Laplace formula (cofactor expansion), Gaussian elimination or other algorithms must be used to calculate the determinant. A= | 1 -2 5 2| | 0 0 3 0| | 2 -4 -3 5| | 2 0 3 5| I figured the easiest way to compute this problem would be to use a cofactor . Mathematics understanding that gets you . If you're struggling to clear up a math equation, try breaking it down into smaller, more manageable pieces. Once you have determined what the problem is, you can begin to work on finding the solution. Find the determinant of the. Gauss elimination is also used to find the determinant by transforming the matrix into a reduced row echelon form by swapping rows or columns, add to row and multiply of another row in order to show a maximum of zeros. Matrix Minors & Cofactors Calculator - Symbolab Matrix Minors & Cofactors Calculator Find the Minors & Cofactors of a matrix step-by-step Matrices Vectors full pad Deal with math problems. Compute the solution of \(Ax=b\) using Cramers rule, where, \[ A = \left(\begin{array}{cc}a&b\\c&d\end{array}\right)\qquad b = \left(\begin{array}{c}1\\2\end{array}\right). \nonumber \]. For example, eliminating x, y, and z from the equations a_1x+a_2y+a_3z = 0 (1) b_1x+b_2y+b_3z . The first minor is the determinant of the matrix cut down from the original matrix Except explicit open source licence (indicated Creative Commons / free), the "Cofactor Matrix" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Cofactor Matrix" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) 2 For each element of the chosen row or column, nd its 995+ Consultants 94% Recurring customers Mathematics is a way of dealing with tasks that require e#xact and precise solutions. 4 Sum the results. It's a Really good app for math if you're not sure of how to do the question, it teaches you how to do the question which is very helpful in my opinion and it's really good if your rushing assignments, just snap a picture and copy down the answers. To determine what the math problem is, you will need to take a close look at the information given and use your problem-solving skills. To calculate Cof(M) C o f ( M) multiply each minor by a 1 1 factor according to the position in the matrix. Again by the transpose property, we have \(\det(A)=\det(A^T)\text{,}\) so expanding cofactors along a row also computes the determinant. Of course, not all matrices have a zero-rich row or column. A cofactor is calculated from the minor of the submatrix. We expand along the fourth column to find, \[ \begin{split} \det(A) \amp= 2\det\left(\begin{array}{ccc}-2&-3&2\\1&3&-2\\-1&6&4\end{array}\right)-5 \det \left(\begin{array}{ccc}2&5&-3\\1&3&-2\\-1&6&4\end{array}\right)\\ \amp\qquad - 0\det(\text{don't care}) + 0\det(\text{don't care}). This is the best app because if you have like math homework and you don't know what's the problem you should download this app called math app because it's a really helpful app to use to help you solve your math problems on your homework or on tests like exam tests math test math quiz and more so I rate it 5/5. Calculate the determinant of the matrix using cofactor expansion along the first row Calculate the determinant of the matrix using cofactor expansion along the first row matrices determinant 2,804 Zeros are a good thing, as they mean there is no contribution from the cofactor there. It is clear from the previous example that \(\eqref{eq:1}\)is a very inefficient way of computing the inverse of a matrix, compared to augmenting by the identity matrix and row reducing, as in SubsectionComputing the Inverse Matrix in Section 3.5. $$ Cof_{i,j} = (-1)^{i+j} \text{Det}(SM_i) $$, $$ M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$, $$ Cof(M) = \begin{bmatrix} d & -c \\ -b & a \end{bmatrix} $$, Example: $$ M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \Rightarrow Cof(M) = \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix} $$, $$ M = \begin{bmatrix} a & b & c \\d & e & f \\ g & h & i \end{bmatrix} $$, $$ Cof(M) = \begin{bmatrix} + \begin{vmatrix} e & f \\ h & i \end{vmatrix} & -\begin{vmatrix} d & f \\ g & i \end{vmatrix} & +\begin{vmatrix} d & e \\ g & h \end{vmatrix} \\ & & \\ -\begin{vmatrix} b & c \\ h & i \end{vmatrix} & +\begin{vmatrix} a & c \\ g & i \end{vmatrix} & -\begin{vmatrix} a & b \\ g & h \end{vmatrix} \\ & & \\ +\begin{vmatrix} b & c \\ e & f \end{vmatrix} & -\begin{vmatrix} a & c \\ d & f \end{vmatrix} & +\begin{vmatrix} a & b \\ d & e \end{vmatrix} \end{bmatrix} $$. To enter a matrix, separate elements with commas and rows with curly braces, brackets or parentheses. A determinant is a property of a square matrix. Math is the study of numbers, shapes, and patterns. Once you know what the problem is, you can solve it using the given information. \end{split} \nonumber \]. \end{align*}, Using the formula for the \(3\times 3\) determinant, we have, \[\det\left(\begin{array}{ccc}2&5&-3\\1&3&-2\\-1&6&4\end{array}\right)=\begin{array}{l}\color{Green}{(2)(3)(4) + (5)(-2)(-1)+(-3)(1)(6)} \\ \color{blue}{\quad -(2)(-2)(6)-(5)(1)(4)-(-3)(3)(-1)}\end{array} =11.\nonumber\], \[ \det(A)= 2(-24)-5(11)=-103. We can calculate det(A) as follows: 1 Pick any row or column. \nonumber \] This is called, For any \(j = 1,2,\ldots,n\text{,}\) we have \[ \det(A) = \sum_{i=1}^n a_{ij}C_{ij} = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj}. The sign factor is -1 if the index of the row that we removed plus the index of the column that we removed is equal to an odd number; otherwise, the sign factor is 1. In this section, we give a recursive formula for the determinant of a matrix, called a cofactor expansion. At the end is a supplementary subsection on Cramers rule and a cofactor formula for the inverse of a matrix. To solve a math problem, you need to figure out what information you have. Mathematics is the study of numbers, shapes and patterns. Cofactor Expansion Calculator. We can calculate det(A) as follows: 1 Pick any row or column. Determinant calculation methods Cofactor expansion (Laplace expansion) Cofactor expansion is used for small matrices because it becomes inefficient for large matrices compared to the matrix decomposition methods. Let is compute the determinant of A = E a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 F by expanding along the first row. Calculate matrix determinant with step-by-step algebra calculator. order now \nonumber \]. Mathematical tasks can be difficult to figure out, but with perseverance and a little bit of help, they can be conquered. In particular: The inverse matrix A-1 is given by the formula: It looks a bit like the Gaussian elimination algorithm and in terms of the number of operations performed. 3. det ( A 1) = 1 / det ( A) = ( det A) 1. \nonumber \], Let us compute (again) the determinant of a general \(2\times2\) matrix, \[ A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right). is called a cofactor expansion across the first row of A A. Theorem: The determinant of an n n n n matrix A A can be computed by a cofactor expansion across any row or down any column. and all data download, script, or API access for "Cofactor Matrix" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app! The calculator will find the matrix of cofactors of the given square matrix, with steps shown. For a 2-by-2 matrix, the determinant is calculated by subtracting the reverse diagonal from the main diagonal, which is known as the Leibniz formula. Figure out mathematic tasks Mathematical tasks can be difficult to figure out, but with perseverance and a little bit of help, they can be conquered. an idea ? Calculate how long my money will last in retirement, Cambridge igcse economics coursebook answers, Convert into improper fraction into mixed fraction, Key features of functions common core algebra 2 worksheet answers, Scientific notation calculator with sig figs. Math Input. The determinant can be viewed as a function whose input is a square matrix and whose output is a number. Let A = [aij] be an n n matrix. Our expert tutors can help you with any subject, any time. Determinant by cofactor expansion calculator - The method of expansion by cofactors Let A be any square matrix. We have several ways of computing determinants: Remember, all methods for computing the determinant yield the same number. Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. To solve a math equation, you need to find the value of the variable that makes the equation true. Then the matrix \(A_i\) looks like this: \[ \left(\begin{array}{cccc}1&0&b_1&0\\0&1&b_2&0\\0&0&b_3&0\\0&0&b_4&1\end{array}\right). Math can be a difficult subject for many people, but there are ways to make it easier. We will also discuss how to find the minor and cofactor of an ele. By performing \(j-1\) column swaps, one can move the \(j\)th column of a matrix to the first column, keeping the other columns in order. Math is all about solving equations and finding the right answer. In Definition 4.1.1 the determinant of matrices of size \(n \le 3\) was defined using simple formulas. Algebra 2 chapter 2 functions equations and graphs answers, Formula to find capacity of water tank in liters, General solution of the differential equation log(dy dx) = 2x+y is. Natural Language Math Input. The only such function is the usual determinant function, by the result that I mentioned in the comment. Determinant by cofactor expansion calculator. Here we explain how to compute the determinant of a matrix using cofactor expansion. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. \nonumber \], Now we expand cofactors along the third row to find, \[ \begin{split} \det\left(\begin{array}{ccc}-\lambda&2&7+2\lambda \\ 3&1-\lambda&2+\lambda(1-\lambda) \\ 0&1&0\end{array}\right)\amp= (-1)^{2+3}\det\left(\begin{array}{cc}-\lambda&7+2\lambda \\ 3&2+\lambda(1-\lambda)\end{array}\right)\\ \amp= -\biggl(-\lambda\bigl(2+\lambda(1-\lambda)\bigr) - 3(7+2\lambda) \biggr) \\ \amp= -\lambda^3 + \lambda^2 + 8\lambda + 21. To solve a math equation, you need to find the value of the variable that makes the equation true. Hot Network. Let \(x = (x_1,x_2,\ldots,x_n)\) be the solution of \(Ax=b\text{,}\) where \(A\) is an invertible \(n\times n\) matrix and \(b\) is a vector in \(\mathbb{R}^n \). We repeat the first two columns on the right, then add the products of the downward diagonals and subtract the products of the upward diagonals: \[\det\left(\begin{array}{ccc}1&3&5\\2&0&-1\\4&-3&1\end{array}\right)=\begin{array}{l}\color{Green}{(1)(0)(1)+(3)(-1)(4)+(5)(2)(-3)} \\ \color{blue}{\quad -(5)(0)(4)-(1)(-1)(-3)-(3)(2)(1)}\end{array} =-51.\nonumber\]. Indeed, it is inconvenient to row reduce in this case, because one cannot be sure whether an entry containing an unknown is a pivot or not. (3) Multiply each cofactor by the associated matrix entry A ij. It is often most efficient to use a combination of several techniques when computing the determinant of a matrix. We can find these determinants using any method we wish; for the sake of illustration, we will expand cofactors on one and use the formula for the \(3\times 3\) determinant on the other. . A determinant is a property of a square matrix. Now we show that \(d(A) = 0\) if \(A\) has two identical rows. The method of expansion by cofactors Let A be any square matrix. Suppose A is an n n matrix with real or complex entries. Then, \[ x_i = \frac{\det(A_i)}{\det(A)}. Please, check our dCode Discord community for help requests!NB: for encrypted messages, test our automatic cipher identifier! We only have to compute one cofactor. Feedback and suggestions are welcome so that dCode offers the best 'Cofactor Matrix' tool for free! Learn more in the adjoint matrix calculator. Then det(Mij) is called the minor of aij. Use the Theorem \(\PageIndex{2}\)to compute \(A^{-1}\text{,}\) where, \[ A = \left(\begin{array}{ccc}1&0&1\\0&1&1\\1&1&0\end{array}\right). This proves that \(\det(A) = d(A)\text{,}\) i.e., that cofactor expansion along the first column computes the determinant. If you're looking for a fun way to teach your kids math, try Decide math. Cofactor expansion calculator can help students to understand the material and improve their grades. Math Workbook. Now we show that cofactor expansion along the \(j\)th column also computes the determinant. Once you have found the key details, you will be able to work out what the problem is and how to solve it. The determinant is determined after several reductions of the matrix to the last row by dividing on a pivot of the diagonal with the formula: The matrix has at least one row or column equal to zero. The determinant is noted Det(SM) Det ( S M) or |SM | | S M | and is also called minor. Expanding cofactors along the \(i\)th row, we see that \(\det(A_i)=b_i\text{,}\) so in this case, \[ x_i = b_i = \det(A_i) = \frac{\det(A_i)}{\det(A)}.