(B) Let $A$ be a square matrix of order $n \times n$. The matrix $C = [c_{ij}]$ is the cofactor matrix of $A$. By the definition of the adjoint matrix

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(B) Let $A$ be a square matrix of order $n \times n$. The matrix $C = [c_{ij}]$ is the cofactor matrix of $A$. By the definition of the adjoint matrix,$adj(A) = C^T$,where $C^T$ is the transpose of the cofactor matrix. We know that $|adj(A)| = |A|^{n-1}$. Since the determinant of a matrix is equal to the determinant of its transpose,we have $|C^T| = |C|$. Therefore,$|C| = |adj(A)| = |A|^{n-1}$. Thus,the correct option is $B$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Minors and Co-factors, Product of determinants

Medium

Let $A = [a_{ij}]_{n \times n}$ be a square matrix and let $c_{ij}$ be the cofactor of $a_{ij}$ in $A$. If $C = [c_{ij}]$,then which of the following is true?

A
$|C| = |A|$
B
$|C| = |A|^{n-1}$
C
$|C| = |A|^{n-2}$
D
None of these

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Class 12

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Chapter

3 and 4 .Determinants and Matrices

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Minors and Co-factors, Product of determinants

Find the minors and cofactors of the elements $a_{11}$ and $a_{21}$ in the determinant $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$.

Easy

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If $A = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ 0 & 3 & -5 \end{bmatrix}$,where $A_{ij}$ is the cofactor of the element $a_{ij}$ of matrix $A$,then $a_{21} A_{21} + a_{22} A_{22} + a_{23} A_{23} = $

MediumMHT CET 2020

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Using cofactors of elements of the second row,evaluate $\Delta = \left|\begin{array}{lll}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{array}\right|$.

Easy

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The sum of the products of the elements of any row of a determinant $A$ with the corresponding co-factors is always equal to

Medium

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The minors of $-4$ and $9$ and the co-factors of $-4$ and $9$ in the determinant $\left| \begin{array}{ccc} -1 & -2 & 3 \\ -4 & -5 & -6 \\ -7 & 8 & 9 \end{array} \right|$ are respectively:

Easy

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Let $A = [a_{ij}]_{n \times n}$ be a square matrix and let $c_{ij}$ be the cofactor of $a_{ij}$ in $A$. If $C = [c_{ij}]$,then which of the following is true?

Similar Questions

Find the minors and cofactors of the elements $a_{11}$ and $a_{21}$ in the determinant $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$.

If $A = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ 0 & 3 & -5 \end{bmatrix}$,where $A_{ij}$ is the cofactor of the element $a_{ij}$ of matrix $A$,then $a_{21} A_{21} + a_{22} A_{22} + a_{23} A_{23} = $

Using cofactors of elements of the second row,evaluate $\Delta = \left|\begin{array}{lll}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{array}\right|$.

The sum of the products of the elements of any row of a determinant $A$ with the corresponding co-factors is always equal to

The minors of $-4$ and $9$ and the co-factors of $-4$ and $9$ in the determinant $\left| \begin{array}{ccc} -1 & -2 & 3 \\ -4 & -5 & -6 \\ -7 & 8 & 9 \end{array} \right|$ are respectively:

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