If Aij is the cofactor of the element aij of the determinant - 2 -3 5- 6 0 4- 1 5 -7 - then write the value of a32-A32-

#10; #10; #10; #9; #10; #9; #10; #9; #10; #9; #10; #10; #10; #10;Given a square matrix, the cofactor of a_ij is denoted by ...

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Standard XII

Mathematics

Cofactor

If Aij is t...

Question

If $A_{i j}$ 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 $a_{32} . A_{32} .$

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Solution

Given a square matrix, the cofactor of $a_{i j}$ is denoted by $A_{i j} = (- 1)^{i + j} M_{i j}$ , where $M_{i j}$ is the minor of the entry $a_{i j}$
$M_{i j} = ∣ ∣ ∣ \begin{matrix} 2 & 5 6 & 4 \end{matrix} ∣ ∣ ∣$

From the matrix, $a_{32} = 5$

Determinant of $A_{32} = (- 1)^{3 + 2} ∣ ∣ ∣ \begin{matrix} 2 & 5 6 & 4 \end{matrix} ∣ ∣ ∣ = - (- 8 - 30) = 22$

Therefore, $a_{32} . A_{32} = 5 \times 22 = 110$ .

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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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If Aij is the cofactor of the element aij of the determinant ⎡⎢⎣2−3560415−7⎤⎥⎦ then write the value of a32.A32.

Given a square matrix, the cofactor of aij is denoted by Aij=(−1)i+jMij, where Mij is the minor of the entry aijMij=∣∣∣2564∣∣∣ From the matrix, a32=5 Determinant of A32=(−1)3+2∣∣∣2564∣∣∣=−(−8−30)=22 Therefore, a32.A32=5×22=110.

If $A_{i j}$ 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 $a_{32} . A_{32} .$