Minors and Cofactors of a Matrix (3×3 and 2×2) with Examples

Here you will learn how to find minors and cofactors of a matrix of order 3×3 and 2×2 with examples.

Let’s begin –

How to find Minors and Cofactors of a Matrix

Minor of Matrix (3×3 and 2×2) 

Let A = \([a_{ij}]\) be a square matrix of order n. The minor \(M_{ij}\) of \(a_{ij}\) in A is the determinant of the square sub-matrix of order (n – 1) obtained by leaving \(i^{ith}\) row and \(j^{ith}\) column of A.

Example : if A = \(\begin{bmatrix} 4 & -7 \\ -3 & 2 \end{bmatrix}\), then

\(M_{11}\) = Minor of  \(a_{11}\) = 2,  \(M_{12}\) = Minor of  \(a_{12}\) = -3, 

\(M_{21}\) = Minor of  \(a_{21}\) = -7,  \(M_{22}\) = Minor of  \(a_{22}\) = 4

If A = \(\begin{bmatrix} 1 & 2 & 3 \\ -3 & 2 & -1 \\ 2 & -4 & 3 \end{bmatrix}\), then

\(M_{11}\) = Minor of  \(a_{11}\)

\(\implies\) \(M_{11}\) = Determinant of the 2×2 square sub matrix obtained by leaving first row and first column of A

\(\implies\) \(M_{11}\) = \(\begin{vmatrix} 2 & -1 \\ -4 & 3 \end{vmatrix}\)

Similarly, we obtain

\(M_{12}\) = Minor of  \(a_{12}\) = \(\begin{vmatrix} -3 & -1 \\  2 & 3 \end{vmatrix}\) = -7

\(M_{13}\) = Minor of  \(a_{13}\) = \(\begin{vmatrix} -3 & 2 \\ 2 & -4 \end{vmatrix}\) = 8

\(M_{21}\) = Minor of  \(a_{21}\) = \(\begin{vmatrix} 2 & 3 \\ -4 & 3 \end{vmatrix}\) = 18

\(M_{22}\) = Minor of  \(a_{22}\) = \(\begin{vmatrix} 1 & 3 \\ 2 & 3 \end{vmatrix}\) = -3    etc.

Cofactor of Matrix (3×3 and 2×2) 

Let A = \([a_{ij}]\) be a square matrix of order n. The cofactor \(C_{ij}\) of \(a_{ij}\) in A is equal to \((-1)^{i+j}\) times the determinant of the sub-matrix of order (n – 1) obtained by leaving \(i^{ith}\) row and \(j^{ith}\) column of A.

It follows from this definition that

\(C_{ij}\) = Cofactor of \(a_{ij}\) in A = \((-1)^{i+j}\)\(M_{ij}\), where \(M_{ij}\) is minor of \(a_{ij}\) in A.

Thus we have

\(C_{ij}\) = \(M_{ij}\) if i + j  is even

\(C_{ij}\) = \(-M_{ij}\) if i + j  is odd

Example : if A = \(\begin{bmatrix} 4 & -7 \\ -3 & 2 \end{bmatrix}\), then

\(C_{11}\) = \((-1)^{1+1}\)\(M_{11}\) = \(M_{11}\)  = 2, \(C_{12}\) = \((-1)^{1+2}\)\(M_{12}\) = -\(M_{12}\)  = -(-3) = 3,

\(C_{21}\) = \((-1)^{2+1}\)\(M_{21}\) = \(M_{21}\)  = -(-7) = 7, \(C_{22}\) = \((-1)^{2+2}\)\(M_{22}\) = \(M_{22}\)  = 4

If A = \(\begin{bmatrix} 1 & 2 & 3 \\ -3 & 2 & -1 \\ 2 & -4 & 3 \end{bmatrix}\), then

\(C_{11}\) = \((-1)^{1+1}\)\(M_{11}\) = \(M_{11}\) = \(\begin{vmatrix} 2 & -1 \\ -4 & 3 \end{vmatrix}\) = 2

\(C_{12}\) = \((-1)^{1+2}\)\(M_{12}\) = -\(M_{12}\) = -\(\begin{vmatrix} -3 & -1 \\ 2 & 3 \end{vmatrix}\) = 7

\(C_{13}\) = \((-1)^{1+3}\)\(M_{13}\) = \(M_{13}\) = \(\begin{vmatrix} -3 & 2 \\ 2 & -4 \end{vmatrix}\) = 8

\(C_{23}\) = \((-1)^{2+3}\)\(M_{23}\) = -\(M_{23}\) = -\(\begin{vmatrix} 1 & 2 \\ 2 & -4 \end{vmatrix}\) = 8    etc.

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