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Adjoint of a Matrix

Find the mino...

Question

Find the minors and cofactors of elements $a_{23}, a_{32} and a_{13}$ of matrix $A = (a_{ij}] = ⎛ ⎜ ⎝ \begin{matrix} 5 & 6 & 7 5 & 2 & 3 4 & 8 & 9 \end{matrix} ⎤ ⎥ ⎦ .$

A

M_{23}, M_{32,} M_{13} = 16, - 20, 32 C_{23}, C_{32,} C_{13} = - 16, 20, 32

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B

M_{23}, M_{32,} M_{13} = 16, - 20, 32 C_{23}, C_{32,} C_{13} = - 16, 20, - 32

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C

M_{23}, M_{32,} M_{13} = 16, - 20, - 32 C_{23}, C_{32,} C_{13} = - 16, 20, 32

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D

M_{23}, M_{32,} M_{13} = 16, - 20, 32 C_{23}, C_{32,} C_{13} = - 16, - 20, 32

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Solution

The correct option is A $M_{23}, M_{32,} M_{13} = 16, - 20, 32 C_{23}, C_{32,} C_{13} = - 16, 20, 32$
$C_{ij} = {(- 1)}^{i + j} M_{ij} M_{23} = (\begin{matrix} 5 & 6 4 & 8 \end{matrix} ∣ ∣ ∣ = 40 - 24 = 16 C_{23} = (- M_{23}) = - 16 M_{13} = (\begin{matrix} 5 & 2 4 & 8 \end{matrix} ∣ ∣ ∣ = 40 - 8 = 32 C_{13} = (M_{13}) = 32 M_{32} = (\begin{matrix} 5 & 7 5 & 3 \end{matrix} ∣ ∣ ∣ = 15 - 35 = - 20 C_{32} = - M_{32} = 20 Here M and C represents Minor and Cofactor respectively .$

option (a) is correct.

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Q. Find the minors and cofactors of elements

a_{23}, a_{32} and a_{13}

A = (a_{ij}] = ⎛ ⎜ ⎝ \begin{matrix} 5 & 6 & 7 5 & 2 & 3 4 & 8 & 9 \end{matrix} ⎤ ⎥ ⎦ .

A_{i j}

is the cofactor of the element

a_{i j}

of the determinant

⎡ ⎢ ⎣ \begin{matrix} 2 & - 3 & 5 6 & 0 & 4 1 & 5 & - 7 \end{matrix} ⎤ ⎥ ⎦

then write the value of

a_{32} . A_{32} .

A = [a_{i j}]

is a matrix of order

2 \times 2

, such that |A| = - 15 and

C_{i j}

represents the cofactor of

a_{i j}

a_{21} C_{21} + a_{22} C_{22} .

Find the cofactor of the element $a_{32}$ in the matrix $⎡ ⎢ ⎣ \begin{matrix} 5 & 1 & 2 3 & 2 & 1 9 & 9 & 5 \end{matrix} ⎤ ⎥ ⎦$

If $Δ ∣ ∣ ∣ ∣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ∣ ∣ ∣ ∣$ and $A_{i j}$ is cofactor of $a_{i j}$ , then value of $Δ$ is given by

a) $a_{11} A_{31} + a_{12} A_{32} + a_{13} A_{33}$
b) $a_{11} A_{11} + a_{12} A_{21} + a_{13} A_{31}$
c) $a_{21} A_{11} + a_{22} A_{12} + a_{23} A_{13}$
d) $a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}$

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