Difference between revisions of "Minors and cofactors"

Revision as of 16:00, 22 May 2014

Overview: Matrices

The minor $M_{ij}(\mathbf{A})$ of an n-by-n square matrix $\mathbf{A}$ is the determinant of a smaller square matrix obtained by removing the row $i$ and the column $j$ from $\mathbf{A}$ .

Multiplying the minor with $(-1)^{i+j}$ results in the cofactor $C_{i,j}(\mathbf{A})$ :

$C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})$

Example: Minors and cofactors of a 3-by-3 matrix

\mathbf{A}_e = \left[\begin{array}{ccc} 1&0&1\\ 3&1&0\\ 1&0&2 \end{array}\right]

The minors $M_{22}(\mathbf{A}_e)$ and $M_{31}(\mathbf{A}_e)$ for example are defined as

M_{22}(\mathbf{A}_e)= \left|\begin{array}{ccc} 1&\Box&1\\ \Box&\Box&\Box\\ 1&\Box&2 \end{array}\right|= \left|\begin{array}{cc} 1&1\\ 1&2 \end{array}\right|=2-1=1

M_{31}(\mathbf{A}_e)= \left|\begin{array}{ccc} \Box & 0&1\\ \Box & 1&0\\ \Box & \Box & \Box\\ \end{array}\right|= \left|\begin{array}{cc} 0&1\\ 1&0 \end{array}\right|=0-1=-1

The corresponding cofactors in that case are

C_{22}(\mathbf{A}_e)=(-1)^{2+2}M_{22}(\mathbf{A}_e)=(-1)^4\cdot1=1

C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1

Example: Minors and cofactors of a 4-by-4 matrix

$\mathbf{A}_e = \left[\begin{array}{cccc} 1 & 2 & 0 & 0\\ 3 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 2 & 1 \end{array}\right]$

The minors $M_{14}(\mathbf{A}_e)$ and $M_{31}(\mathbf{A}_e)$ for example are defined as

$M_{14}(\mathbf{A}_e)= \left|\begin{array}{cccc} \Box & \Box & \Box & \Box\\ 3 & 0 & 1 & \Box\\ 0 & 1 & 0 & \Box\\ 0 & 0 & 2 & \Box \end{array}\right|= \left|\begin{array}{ccc} 3 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{array}\right|=6-0=6$
$M_{31}(\mathbf{A}_e)= \left|\begin{array}{cccc} \Box & 2 & 0 & 0\\ \Box & 0 & 1 & 1\\ \Box & \Box & \Box & \Box\\ \Box & 0 & 2 & 1 \end{array}\right|= \left|\begin{array}{ccc} 2 & 0 & 0\\ 0 & 1 & 1\\ 0 & 2 & 1 \end{array}\right|=2-4=-2$

The corresponding cofactors in that case are

$C_{14}(\mathbf{A}_e)=(-1)^{1+4}M_{14}(\mathbf{A}_e)=(-1)^5\cdot6=-6$

$C_{31}(\mathbf{A}_e)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-2=-2$