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Adjoint Matrix Calculator takes matrix as input and finds its adjoint matrix, with detailed step by step calculations.

How to use Adjoint Matrix Calculator?

1. Find the Adjoint matrix of the following matrix A.

\( A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\9 & 8 & 7\end{bmatrix} \)

Solution

Since the given matrix is of size 3×3, follow these steps with the Adjoint Matrix Calculator.

Enter value of 3 in input fields: m=3 and n=3.
A matrix with specified size of 3×3 appears, with input field for each element in the matrix.
Enter the given matrix values, and click on Calculate button.
A step-by-step solution with the following steps will be displayed.

Given:

Matrix \( A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix} \)

To find the Adjoint Matrix, we shall first compute Cofactor Matrix, and then Transpose it.

Computing Cofactor Matrix

The formula to find the cofactor of each element \(a_{ij}\) in matrix A, where \(a_{ij}\) is the element at i^th row, and j^th column is

Cofactor of \(a_{ij} = (-1)^{i+j}\cdot det(minor(a_{ij}))\)

where

\(minor(a_{ij})\) is the minor matrix of element \(a_{ij}\)
\(det(minor(a_{ij}))\) is the determinant of minor matrix of element \(a_{ij}\)

Cofactors of elements in row 1

Cofactor of \(a_{11} = (-1)^{1+1}\cdot |minor(a_{11})|\) \( = (-1)^{2}\cdot \begin{vmatrix}5 & 6\\8 & 9\end{vmatrix} \)\( = -3 \)

Cofactor of \(a_{12} = (-1)^{1+2}\cdot |minor(a_{12})|\) \( = (-1)^{3}\cdot \begin{vmatrix}4 & 6\\7 & 9\end{vmatrix} \)\( = 6 \)

Cofactor of \(a_{13} = (-1)^{1+3}\cdot |minor(a_{13})|\) \( = (-1)^{4}\cdot \begin{vmatrix}4 & 5\\7 & 8\end{vmatrix} \)\( = -3 \)

Cofactors of elements in row 2

Cofactor of \(a_{21} = (-1)^{2+1}\cdot |minor(a_{21})|\) \( = (-1)^{3}\cdot \begin{vmatrix}2 & 3\\8 & 9\end{vmatrix} \)\( = 6 \)

Cofactor of \(a_{22} = (-1)^{2+2}\cdot |minor(a_{22})|\) \( = (-1)^{4}\cdot \begin{vmatrix}1 & 3\\7 & 9\end{vmatrix} \)\( = -12 \)

Cofactor of \(a_{23} = (-1)^{2+3}\cdot |minor(a_{23})|\) \( = (-1)^{5}\cdot \begin{vmatrix}1 & 2\\7 & 8\end{vmatrix} \)\( = 6 \)

Cofactors of elements in row 3

Cofactor of \(a_{31} = (-1)^{3+1}\cdot |minor(a_{31})|\) \( = (-1)^{4}\cdot \begin{vmatrix}2 & 3\\5 & 6\end{vmatrix} \)\( = -3 \)

Cofactor of \(a_{32} = (-1)^{3+2}\cdot |minor(a_{32})|\) \( = (-1)^{5}\cdot \begin{vmatrix}1 & 3\\4 & 6\end{vmatrix} \)\( = 6 \)

Cofactor of \(a_{33} = (-1)^{3+3}\cdot |minor(a_{33})|\) \( = (-1)^{6}\cdot \begin{vmatrix}1 & 2\\4 & 5\end{vmatrix} \)\( = -3 \)

Consolidate all the cofactors computed for elements in the matrix A to get the Cofactor Matrix of A.

Cofactor Matrix of A \( = \begin{bmatrix}-3 & 6 & -3\\6 & -12 & 6\\-3 & 6 & -3\end{bmatrix} \)

Transpose Cofactor Matrix

Transform rows to columns,

Adjoint Matrix of A \( = \begin{bmatrix}-3 & 6 & -3\\6 & -12 & 6\\-3 & 6 & -3\end{bmatrix} ^T\)

\( = \begin{bmatrix}-3 & 6 & -3\\6 & -12 & 6\\-3 & 6 & -3\end{bmatrix} \)

Result

Therefore, the adjoint matrix of given matrix A, is

Adjoint Matrix of \( A = \begin{bmatrix}-3 & 6 & -3\\6 & -12 & 6\\-3 & 6 & -3\end{bmatrix} \)

2. Find the Adjoint matrix of the following matrix A.

\( A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} \)

Solution

Since the given matrix is of size 2×2, follow these steps with the Adjoint Matrix Calculator.

Enter value of 2 in input fields: m=2 and n=2.
A matrix with specified size of 2×2 appears, with input field for each element in the matrix.
Enter the given matrix values, and click on Calculate button.
A step-by-step solution with the following steps will be displayed.

Given:

Matrix \( A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} \)

To find the Adjoint Matrix, we shall first compute Cofactor Matrix, and then Transpose it.

Computing Cofactor Matrix

The formula to find the cofactor of each element \(a_{ij}\) in matrix A, where \(a_{ij}\) is the element at i^th row, and j^th column is

Cofactor of \(a_{ij} = (-1)^{i+j}\cdot det(minor(a_{ij}))\)

where

\(minor(a_{ij})\) is the minor matrix of element \(a_{ij}\)
\(det(minor(a_{ij}))\) is the determinant of minor matrix of element \(a_{ij}\)

Cofactors of elements in row 1

Cofactor of \(a_{11} = (-1)^{1+1}\cdot |minor(a_{11})|\) \( = (-1)^{2}\cdot \begin{vmatrix}4\end{vmatrix} \)\( = 4 \)

Cofactor of \(a_{12} = (-1)^{1+2}\cdot |minor(a_{12})|\) \( = (-1)^{3}\cdot \begin{vmatrix}3\end{vmatrix} \)\( = -3 \)

Cofactors of elements in row 2

Cofactor of \(a_{21} = (-1)^{2+1}\cdot |minor(a_{21})|\) \( = (-1)^{3}\cdot \begin{vmatrix}2\end{vmatrix} \)\( = -2 \)

Cofactor of \(a_{22} = (-1)^{2+2}\cdot |minor(a_{22})|\) \( = (-1)^{4}\cdot \begin{vmatrix}1\end{vmatrix} \)\( = 1 \)

Consolidate all the cofactors computed for elements in the matrix A to get the Cofactor Matrix of A.

Cofactor Matrix of A \( = \begin{bmatrix}4 & -3\\-2 & 1\end{bmatrix} \)

Transpose Cofactor Matrix

Transform rows to columns,

Adjoint Matrix of A \( = \begin{bmatrix}4 & -3\\-2 & 1\end{bmatrix} ^T\)

\( = \begin{bmatrix}4 & -2\\-3 & 1\end{bmatrix} \)

Result

Therefore, the adjoint matrix of given matrix A, is

Adjoint Matrix of \( A = \begin{bmatrix}4 & -2\\-3 & 1\end{bmatrix} \)

Matrix Calculators

Matrix Addition Calculator Matrix Subtraction Calculator Matrix Multiplication Calculator Matrix Transpose Calculator Matrix Inverse Calculator Matrix Determinant Calculator Matrix Rank Calculator Cofactor Matrix Calculator Adjoint Matrix Calculator

{ "topic": "matrix-adjoint", "title": "Adjoint Matrix Calculator", "tag": "Matrix", "template": "matrix", "n_matrices": 1, "different_size": false, "symbol": "", "description": "Adjoint Matrix Calculator takes matrix as input and finds its adjoint matrix, with detailed step by step calculations.", "content": "<div class=\"flex_row\"><div class=\"input_section\"><form onsubmit=\"return calculate()\"><div style=\"padding:10px\"><h3>Enter Matrix Size (m × m)</h3><div id=\"matrix_size\"><input type=\"number\" required size=\"1\" step=\"1\" min=\"1\" id=\"a_m\" name=\"a_m\" placeholder=\"m\" /> × <input type=\"number\" required step=\"1\" step=\"1\" min=\"1\" id=\"a_n\" name=\"a_n\" placeholder=\"m\" /></div><p class=\"hint\">square matrix of <strong>m</strong> rows and <strong>m</strong> columns</p><h3>Enter matrix <strong>A</strong></h3><div id=\"m_a\"><span class=\"input_note\"></span></div></div><button type=\"submit\" class=\"calculate primary_action\"\">Calculate</button></form></div></div><div id=\"solution\"></div><h2>How to use Adjoint Matrix Calculator?</h2><h3 class=\"example\"><span class=\"example_number\">1.</span> Find the Adjoint matrix of the following matrix A.</h3><p class=\"step\">\\( A = \\begin{bmatrix}1 & 2 & 3\\\\4 & 5 & 6\\\\9 & 8 & 7\\end{bmatrix} \\)</p><h3 class=\"solution\">Solution</h3><div class=\"solution\"><p>Since the given matrix is of size 3×3, follow these steps with the Adjoint Matrix Calculator.<ol><li>Enter value of 3 in input fields: <span class=\"strong\">m=3</span> and <span class=\"strong\">n=3</span>.</li><li>A matrix with specified size of 3×3 appears, with input field for each element in the matrix.</li><li>Enter the given matrix values, and click on Calculate button.</li><li>A step-by-step solution with the following steps will be displayed.</li></ol></p><p><strong>Given:</strong></p><p class=\"step\">Matrix \\( A = \\begin{bmatrix}1 & 2 & 3\\\\4 & 5 & 6\\\\7 & 8 & 9\\end{bmatrix} \\)</p><p>To find the Adjoint Matrix, we shall first compute Cofactor Matrix, and then Transpose it.</p><p class=\"strong\">Computing Cofactor Matrix</p><p>The formula to find the cofactor of each element \\(a_{ij}\\) in matrix A, where \\(a_{ij}\\) is the element at i<sup>th</sup> row, and j<sup>th</sup> column is</p><p class=\"plain board color1\">Cofactor of \\(a_{ij} = (-1)^{i+j}\\cdot det(minor(a_{ij}))\\)</p><p>where</p><ul><li>\\(minor(a_{ij})\\) is the minor matrix of element \\(a_{ij}\\)</li><li>\\(det(minor(a_{ij}))\\) is the determinant of minor matrix of element \\(a_{ij}\\)</li></ul><p class=\"strong\">Cofactors of elements in row 1</p><p class=\"step\">Cofactor of \\(a_{11} = (-1)^{1+1}\\cdot |minor(a_{11})|\\) \\( = (-1)^{2}\\cdot \\begin{vmatrix}5 & 6\\\\8 & 9\\end{vmatrix} \\)\\( = -3 \\)</p><p class=\"step\">Cofactor of \\(a_{12} = (-1)^{1+2}\\cdot |minor(a_{12})|\\) \\( = (-1)^{3}\\cdot \\begin{vmatrix}4 & 6\\\\7 & 9\\end{vmatrix} \\)\\( = 6 \\)</p><p class=\"step\">Cofactor of \\(a_{13} = (-1)^{1+3}\\cdot |minor(a_{13})|\\) \\( = (-1)^{4}\\cdot \\begin{vmatrix}4 & 5\\\\7 & 8\\end{vmatrix} \\)\\( = -3 \\)</p><p class=\"strong\">Cofactors of elements in row 2</p><p class=\"step\">Cofactor of \\(a_{21} = (-1)^{2+1}\\cdot |minor(a_{21})|\\) \\( = (-1)^{3}\\cdot \\begin{vmatrix}2 & 3\\\\8 & 9\\end{vmatrix} \\)\\( = 6 \\)</p><p class=\"step\">Cofactor of \\(a_{22} = (-1)^{2+2}\\cdot |minor(a_{22})|\\) \\( = (-1)^{4}\\cdot \\begin{vmatrix}1 & 3\\\\7 & 9\\end{vmatrix} \\)\\( = -12 \\)</p><p class=\"step\">Cofactor of \\(a_{23} = (-1)^{2+3}\\cdot |minor(a_{23})|\\) \\( = (-1)^{5}\\cdot \\begin{vmatrix}1 & 2\\\\7 & 8\\end{vmatrix} \\)\\( = 6 \\)</p><p class=\"strong\">Cofactors of elements in row 3</p><p class=\"step\">Cofactor of \\(a_{31} = (-1)^{3+1}\\cdot |minor(a_{31})|\\) \\( = (-1)^{4}\\cdot \\begin{vmatrix}2 & 3\\\\5 & 6\\end{vmatrix} \\)\\( = -3 \\)</p><p class=\"step\">Cofactor of \\(a_{32} = (-1)^{3+2}\\cdot |minor(a_{32})|\\) \\( = (-1)^{5}\\cdot \\begin{vmatrix}1 & 3\\\\4 & 6\\end{vmatrix} \\)\\( = 6 \\)</p><p class=\"step\">Cofactor of \\(a_{33} = (-1)^{3+3}\\cdot |minor(a_{33})|\\) \\( = (-1)^{6}\\cdot \\begin{vmatrix}1 & 2\\\\4 & 5\\end{vmatrix} \\)\\( = -3 \\)</p><p>Consolidate all the cofactors computed for elements in the matrix A to get the Cofactor Matrix of A.</p><p class=\"plain board color1\">Cofactor Matrix of A \\( = \\begin{bmatrix}-3 & 6 & -3\\\\6 & -12 & 6\\\\-3 & 6 & -3\\end{bmatrix} \\)</p><p class=\"strong\">Transpose Cofactor Matrix</p><p>Transform rows to columns,</p><p class=\"step\">Adjoint Matrix of A \\( = \\begin{bmatrix}-3 & 6 & -3\\\\6 & -12 & 6\\\\-3 & 6 & -3\\end{bmatrix} ^T\\)</p><p class=\"step\"> \\( = \\begin{bmatrix}-3 & 6 & -3\\\\6 & -12 & 6\\\\-3 & 6 & -3\\end{bmatrix} \\)</p><p class=\"strong\">Result</p><p>Therefore, the adjoint matrix of given matrix A, is</p><p class=\"board color1\">Adjoint Matrix of \\( A = \\begin{bmatrix}-3 & 6 & -3\\\\6 & -12 & 6\\\\-3 & 6 & -3\\end{bmatrix} \\)</p></div><h3 class=\"example\"><span class=\"example_number\">2.</span> Find the Adjoint matrix of the following matrix A.</h3><p class=\"step\">\\( A = \\begin{bmatrix}1 & 2\\\\3 & 4\\end{bmatrix} \\)</p><h3 class=\"solution\">Solution</h3><div class=\"solution\"><p>Since the given matrix is of size 2×2, follow these steps with the Adjoint Matrix Calculator.<ol><li>Enter value of 2 in input fields: <span class=\"strong\">m=2</span> and <span class=\"strong\">n=2</span>.</li><li>A matrix with specified size of 2×2 appears, with input field for each element in the matrix.</li><li>Enter the given matrix values, and click on Calculate button.</li><li>A step-by-step solution with the following steps will be displayed.</li></ol></p><p><strong>Given:</strong></p><p class=\"step\">Matrix \\( A = \\begin{bmatrix}1 & 2\\\\3 & 4\\end{bmatrix} \\)</p><p>To find the Adjoint Matrix, we shall first compute Cofactor Matrix, and then Transpose it.</p><p class=\"strong\">Computing Cofactor Matrix</p><p>The formula to find the cofactor of each element \\(a_{ij}\\) in matrix A, where \\(a_{ij}\\) is the element at i<sup>th</sup> row, and j<sup>th</sup> column is</p><p class=\"plain board color1\">Cofactor of \\(a_{ij} = (-1)^{i+j}\\cdot det(minor(a_{ij}))\\)</p><p>where</p><ul><li>\\(minor(a_{ij})\\) is the minor matrix of element \\(a_{ij}\\)</li><li>\\(det(minor(a_{ij}))\\) is the determinant of minor matrix of element \\(a_{ij}\\)</li></ul><p class=\"strong\">Cofactors of elements in row 1</p><p class=\"step\">Cofactor of \\(a_{11} = (-1)^{1+1}\\cdot |minor(a_{11})|\\) \\( = (-1)^{2}\\cdot \\begin{vmatrix}4\\end{vmatrix} \\)\\( = 4 \\)</p><p class=\"step\">Cofactor of \\(a_{12} = (-1)^{1+2}\\cdot |minor(a_{12})|\\) \\( = (-1)^{3}\\cdot \\begin{vmatrix}3\\end{vmatrix} \\)\\( = -3 \\)</p><p class=\"strong\">Cofactors of elements in row 2</p><p class=\"step\">Cofactor of \\(a_{21} = (-1)^{2+1}\\cdot |minor(a_{21})|\\) \\( = (-1)^{3}\\cdot \\begin{vmatrix}2\\end{vmatrix} \\)\\( = -2 \\)</p><p class=\"step\">Cofactor of \\(a_{22} = (-1)^{2+2}\\cdot |minor(a_{22})|\\) \\( = (-1)^{4}\\cdot \\begin{vmatrix}1\\end{vmatrix} \\)\\( = 1 \\)</p><p>Consolidate all the cofactors computed for elements in the matrix A to get the Cofactor Matrix of A.</p><p class=\"plain board color1\">Cofactor Matrix of A \\( = \\begin{bmatrix}4 & -3\\\\-2 & 1\\end{bmatrix} \\)</p><p class=\"strong\">Transpose Cofactor Matrix</p><p>Transform rows to columns,</p><p class=\"step\">Adjoint Matrix of A \\( = \\begin{bmatrix}4 & -3\\\\-2 & 1\\end{bmatrix} ^T\\)</p><p class=\"step\"> \\( = \\begin{bmatrix}4 & -2\\\\-3 & 1\\end{bmatrix} \\)</p><p class=\"strong\">Result</p><p>Therefore, the adjoint matrix of given matrix A, is</p><p class=\"board color1\">Adjoint Matrix of \\( A = \\begin{bmatrix}4 & -2\\\\-3 & 1\\end{bmatrix} \\)</p></div>", "links": [ [ "Matrix Addition Calculator", "/calculators/algebra/matrices/matrix-addition" ], [ "Matrix Subtraction Calculator", "/calculators/algebra/matrices/matrix-subtraction" ], [ "Matrix Multiplication Calculator", "/calculators/algebra/matrices/matrix-multiplication" ], [ "Matrix Transpose Calculator", "/calculators/algebra/matrices/matrix-transpose" ], [ "Matrix Inverse Calculator", "/calculators/algebra/matrices/matrix-inverse" ], [ "Matrix Determinant Calculator", "/calculators/algebra/matrices/matrix-determinant" ], [ "Matrix Rank Calculator", "/calculators/algebra/matrices/matrix-rank" ], [ "Cofactor Matrix Calculator", "/calculators/algebra/matrices/matrix-cofactor" ], [ "Adjoint Matrix Calculator", "/calculators/algebra/matrices/matrix-adjoint" ] ] }

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Adjoint Matrix Calculator

Enter Matrix Size (m × m)

Enter matrix A

How to use Adjoint Matrix Calculator?

1. Find the Adjoint matrix of the following matrix A.

Solution

2. Find the Adjoint matrix of the following matrix A.

Solution

Matrix Calculators