Question

Find the adjoint of the following matrices.

$[\begin{matrix} 1 & - 1 & 2 \\ - 2 & 3 & 5 \\ - 2 & 0 & - 1 \end{matrix}]$

Accepted Answer

Given $[\begin{matrix} 1 & - 1 & 2 \\ - 2 & 3 & 5 \\ - 2 & 0 & - 1 \end{matrix}]$ $[\begin{matrix} 1 & - 1 & 2 \\ - 2 & 3 & 5 \\ - 2 & 0 & - 1 \end{matrix}] = |\begin{array}{l} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}|$

Minors of elements are:

$M_{11} = [\begin{matrix} 3 & 5 \\ 0 & - 1 \end{matrix}] = - 3 - 0 = - 3$

$M_{12} = [\begin{matrix} - 2 & 5 \\ - 2 & - 1 \end{matrix}] = 2 + 10 = 12$

$M_{13} = [\begin{matrix} - 2 & 3 \\ - 2 & 0 \end{matrix}] = 0 + 6 = 6$

$M_{21} = [\begin{matrix} - 1 & 2 \\ 0 & - 1 \end{matrix}] = 1 - 0 = 1$

$M_{22} = [\begin{matrix} 1 & 2 \\ - 2 & - 1 \end{matrix}] = - 1 + 4 = 3$

$M_{23} = [\begin{matrix} 1 & - 1 \\ - 2 & 0 \end{matrix}] = 0 - 2 = - 2$

$M_{31} = [\begin{matrix} - 1 & 2 \\ 3 & 5 \end{matrix}] = - 5 - 6 = - 11$

$M_{32} = [\begin{matrix} 1 & 2 \\ - 2 & 5 \end{matrix}] = 5 + 4 = 9$

$M_{33} = [\begin{matrix} 1 & - 1 \\ - 2 & 3 \end{matrix}] = 3 - 2 = 1$

Co-factors are:

$c_{11} = (- 1)^{1 + 1} M_{11} = - 3 \times (- 1)^{2} = - 3$

$c_{12} = (- 1)^{1 + 2} M_{12} = (- 1)^{3} (12) = - 12$

$C_{13} = (- 1)^{1 + 3} M_{13} = 6$

$c_{21} = (- 1)^{2 + 1} M_{21} = - 1 \times 1 = - 1$

$c_{22} = (- 1)^{2 + 2} M_{22} = 1 \times (3) = 3$

$C_{23} = (- 1)^{2 + 3} M_{23} = - 1 \times (- 2) = 2$

$C_{31} = (- 1)^{3 + 1} M_{31} = 1 \times (- 11) = - 11$

$C_{32} = (- 1)^{3 + 2} M_{32} = - 1 \times 9 = - 9$

$C_{33} = (- 1)^{3 + 3} M_{33} = 1 \times 1 = 1$

Co-factor matrix $= [\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}] = [\begin{matrix} - 3 & - 12 & 6 \\ - 1 & 3 & 2 \\ - 11 & - 9 & 1 \end{matrix}]$

Hence $a d j A = [\begin{matrix} - 3 & - 1 & - 11 \\ - 12 & 3 & - 9 \\ 6 & 2 & 1 \end{matrix}]$

Maharashtra Board Solutions for Chapter: Matrices, Exercise 2: Exercise 2.2

Maharashtra Board Mathematics Solutions for Exercise - Maharashtra Board Solutions for Chapter: Matrices, Exercise 2: Exercise 2.2

Questions from Maharashtra Board Solutions for Chapter: Matrices, Exercise 2: Exercise 2.2 with Hints & Solutions

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