Minor and Cofactor of a Matrix in machine learning

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Minor and Cofactor of a Matrix

Let A be a square matrix of order n. The matrix obtained from A by deleting its ith row and jth column is again a matrix [katex]M_{ij}[/katex] of order n-1 . [katex]M_{ij}[/katex] is called the ijth minor of [katex]A^1[/katex]. Let [katex]M_{i j}[/katex] be the ijth minor of a square matrix A of order n. Then [katex]A_{ij}=(-1)^{i+j} det M_{i j}[/katex] is called the ijth cofactor of A.

Earn Money online without investment Earn Money online from home without investment Observe that the sign on the right hand side of the above equality is positive or negative according as i+j is even of odd. Example 1. Let

A=\left[\begin{array}{rrrr}
2 & -1 & 3 & 4 \\
3 & 2 & 5 & -2 \\
3 & 4 & 2 & 1 \\
-3 & 2 & 5 & 1
\end{array}\right]

Then

M_{23}=\left[\begin{array}{rrr}
2 & -1 & 4 \\
3 & 4 & 1 \\
-3 & 2 & 1
\end{array}\right], \quad M_{42}=\left[\begin{array}{rrr}
2 & 3 & 4 \\
3 & 5 & -2 \\
3 & 2 & 1
\end{array}\right]

and

A_{23}=(-1)^{2+3} det M_{21}=-82\\
A_{42}=(-1)^{4+2} det M_{42}=-45

Consider the expansion of det A . Using the fact that

A_{11}=(-1)^{1+1}\left|\begin{array}{rrr}
a_{22} & a_{23}  \\
a_{32} & a_{33} \\
\end{array}\right|,
A_{12}=(-1)^{1+2}\left|\begin{array}{rrr}
a_{21} & a_{23}  \\
a_{31} & a_{33} \\
\end{array}\right|,
A_{13}=(-1)^{1+3}\left|\begin{array}{rrr}
a_{21} & a_{22}  \\
a_{31} & a_{32} \\
\end{array}\right|,

We may write as

det A=a_{11} A_{11} +a_{12} A_{12}+a_{13} A_{13}

We now define det A of a matrix A of order n as follows: (5.5) Definition. For the matrix

A=\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n} \\
a_{21} & a_{22} & \cdots & a_{2 n} \\
\vdots & \vdots & \cdots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n}
\end{array}\right]

of order n, we define det A by [katex] det A=a_{i1} A_{i1}+a_{i2} A_{i2}+\cdots+a_{in} A_{in} [/katex] The expression on the right hand side of (1) is an expansion of det A by cofactors of the ith row of det A. This is called Laplace’s Expansion of a determinant of order n. The definition (1) given above indicates that the determinant of a miatrix can be cvaluated by any row of the matrix. We only have to be careful about the sign of the cofactor of the corresponding element. Example 2. Let

A=\left[\begin{array}{rrrr}
3 & 2 & 1 & -1 \\
4 & 5 & 1 & 2 \\
-2 & 3 & 0 & 1 \\
2 & 1 & 3 & 5
\end{array}\right] \text {. Then }
=3\left|\begin{array}{rrr}5 & 1 & 2 \\ 3 & 0 & 1 \\ 1 & 3 & 5\end{array}\right|-2\left|\begin{array}{rrr}4 & 1 & 2 \\ -2 & 0 & 1 \\ 2 & 3 & 5\end{array}\right|+1\left|\begin{array}{rrr}4 & 5 & 2 \\ -2 & 3 & 1 \\ 2 & 1 & 5\end{array}\right|
+1\left|\begin{array}{rrr}4 & 5 & 1 \\ -2 & 3 & 0 \\ 2 & 1 & 3\end{array}\right|
=3[5(-3)-1(14)+2(9)]-2[4(-3)-1(-12)+2(-6)]
+1[4(14)-5(-12)+2(-8)]+1[4(9)-5(-6)+1(-8)]
=3(-15-14+18)-2(-12+12-12)+1(56+60-16)
+1(36+30-8)
=-33+24+100+58=149

Remark. Although the above technique to evaluate determinant of an [katex]n \times n[/katex] matrix seems quite straightforward, yet, in practice, it is very laborious to work with when n>3 because it involves a lot of calculations. We shall later discuss methods which simplify these calculations.

Note that, for some special type of matrices, their determinants can be easily evaluated. For instance, if a matrix A is triangular (upper or lower), that is, all its entries below or above the main diagonal are zero respectively, then its determinant is just the product of the elements on the main diagnnal Example 3. Let [katex] A=\left[\begin{array}{llll} a_{11} & 0 & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 \\ a_{31} & a_{32} & a_{13} & 0 \\ a_{41} & a_{62} & a_{43} & a_{44} \end{array}\right] [/katex]

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