(D) The value of a determinant is the sum of the products of elements of any row (or column) with their corresponding co-factors. Thus,$a_1 A_1 + b_1

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$a_1 A_2 + b_1 B_2 + c_1 C_2 = \Delta$

(D) The value of a determinant is the sum of the products of elements of any row (or column) with their corresponding co-factors. Thus,$a_1 A_1 + b_1 B_1 + c_1 C_1 = \Delta$,$a_2 A_2 + b_2 B_2 + c_2 C_2 = \Delta$,and $a_3 A_3 + b_3 B_3 + c_3 C_3 = \Delta$. However,the sum of the products of elements of any row (or column) with the co-factors of a different row (or column) is always zero. Therefore,$a_1 A_2 + b_1 B_2 + c_1 C_2 = 0$,not $\Delta$. Hence,the relation in option $(d)$ is incorrect.

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

Easy

If in the determinant $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$,$A_1, B_1, C_1$ etc. are the co-factors of $a_1, b_1, c_1$ etc.,then which of the following relations is incorrect?

A
$a_1 A_1 + b_1 B_1 + c_1 C_1 = \Delta$
B
$a_2 A_2 + b_2 B_2 + c_2 C_2 = \Delta$
C
$a_3 A_3 + b_3 B_3 + c_3 C_3 = \Delta$
D
$a_1 A_2 + b_1 B_2 + c_1 C_2 = \Delta$

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3 and 4 .Determinants and Matrices

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Minors and Co-factors, Product of determinants

Consider the matrices $A=\begin{bmatrix} x & y & 0 \\ -3 & 1 & 2 \\ 1 & -2 & z \end{bmatrix}$ and $B=\begin{bmatrix} 1 & -2 & -2 \\ 2 & 0 & 1 \\ 2 & 1 & 0 \end{bmatrix}$. If the cofactors of the elements $z$,$1$ (in the $3^{rd}$ row,$2^{nd}$ column),and $x$ of $A$ are $9, 4, 3$ respectively,then $AB=$

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If $A = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$ and $A_{ij}$ is the cofactor of $a_{ij}$,then the value of $a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23}$ is

EasyMHT CET 2023

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Let $A = \begin{bmatrix} 3 & 1 & 2 \\ 1 & 2 & 0 \\ 1 & 1 & 4 \end{bmatrix}$ and $pC_{11} + 4C_{21} - 5C_{32} = -2$, where $C_{ij}$ denotes the cofactor of an element $a_{ij}$ of matrix $A$. Find the value of $p$.

DifficultMHT CET 2026

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If the cofactors of the elements $3$,$7$ and $6$ of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6 \end{bmatrix}$ are $a$,$b$ and $c$ respectively,then $\begin{bmatrix} a & b & c \end{bmatrix} \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix} + \begin{bmatrix} a & b & c \end{bmatrix} \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix} = $

MediumAP EAMCET 2023

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Find the minors and cofactors of the elements of the determinant $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ and verify that $a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}=0$.

Medium

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If in the determinant $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$,$A_1, B_1, C_1$ etc. are the co-factors of $a_1, b_1, c_1$ etc.,then which of the following relations is incorrect?

Similar Questions

If $A = \begin{bmatrix} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$ and $A_{ij}$ is the cofactor of $a_{ij}$,then the value of $a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23}$ is

Let $A = \begin{bmatrix} 3 & 1 & 2 \\ 1 & 2 & 0 \\ 1 & 1 & 4 \end{bmatrix}$ and $pC_{11} + 4C_{21} - 5C_{32} = -2$, where $C_{ij}$ denotes the cofactor of an element $a_{ij}$ of matrix $A$. Find the value of $p$.

Find the minors and cofactors of the elements of the determinant $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ and verify that $a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}=0$.

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