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Q)

Find the adjoint of the following matrices $\begin{bmatrix} 3 & -1 \\2 & -4& \end{bmatrix}$

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A)
Toolbox:
  • Let A = $ [ a_{ij} ] $ be a square matrix of order x. Let $ A_{ij}$ be the cofactor of $ a _{ij}$. Then $ [ A_{ij} ] $ is the matrix of cofactors and $ adj\: A$ ( or adjoint of the matrix A ) is given by $ adj\: A = [ A_{ij} ]^T$
  • Let A be a square matrix of order 2. $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Then $ adj A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ ( shorcut for finding the adjoint )
Step 1
Let A =$ \begin{bmatrix} 3 & -1 \\ 2 & -4 \end{bmatrix}$
$ [ A_{ij} ]= \begin{bmatrix} -4 & -2 \\ 1 & 3 \end{bmatrix}$
Step 2
$ adjA = [ A_{ij} ]^T = \begin{bmatrix} -4 & 1 \\ -2 & 3 \end{bmatrix}$
or by shortcut method: If $ A = \begin{bmatrix} 3 & -1 \\ 2 & -4 \end{bmatrix}$ $ \therefore adjA = \begin{bmatrix} -4 & 1 \\ -2 & 3 \end{bmatrix}$
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