# Minors of a Matrix – Explained with Examples

-

The minors of a matrix are quantities that are obtained by deleting some rows and columns from a matrix and finding the determinant of the resulting submatrix. The minor  M_{ij} is the determinant of the submatrix formed by deleting the i^{th} row and the j^{th} column from the given matrix.

#### Example 1:

Consider the matrix A = \begin{bmatrix} 7 & -2 & 4\\ -8 & 6 & 3\\ 5 & 0 & -1 \end{bmatrix}.

We find all the minors of this matrix by calculating the determinants of the submatrices below.

M_{11} = \left|\begin{array}{cc}6 & 3\\0 & -1\end{array}\right| = (6)\cdot (-1) - (3)\cdot (0) = -6 M_{12} = \left|\begin{array}{cc}-8 & 3\\5 & -1\end{array}\right| = (-8)\cdot (-1) - (3)\cdot (5) = -7 M_{13} = \left|\begin{array}{cc}-8 & 6\\5 & 0\end{array}\right| = -30 M_{21} = \left|\begin{array}{cc}-2 & 4\\0 & -1\end{array}\right| = 2 M_{22} = \left|\begin{array}{cc}7 & 4\\5 & -1\end{array}\right| = -27 M_{23} = \left|\begin{array}{cc}7 & -2\\5 & 0\end{array}\right| = 10 M_{31} = \left|\begin{array}{cc}-2 & 4\\6 & 3\end{array}\right| = -30 M_{32} = \left|\begin{array}{cc}7 & 4\\-8 & 3\end{array}\right| = 53 M_{33} = \left|\begin{array}{cc}7 & -2\\-8 & 6\end{array}\right| = 26

#### Example 2:

We can similarly compute minors for a 4 \times 4 matrix. If we are given the matrix,

A = \begin{bmatrix} 4 & 9 & 5 & 3 \\ -2 & -3 & 8 & 6 \\ 0 & 0 & -4 & 8 \\ 8 & 1 & 0 & 2 \end{bmatrix}

we can similarly compute the minors M_{ij} by deleting the i^{th} row and the j^{th} column and computing the determinant of the resulting submatrix. For example, we can compute M_{23} as,

M_{23} = det \left|\begin{array}{ccc}4 & 9 & 3\\0 & 0 & 8\\8 & 1 & 2\end{array}\right| = 544

The other minors are computed in a similar fashion.

#### Principal Minors of a Matrix:

When we compute minors of a matrix we delete some rows and columns. We say that a minor is principal if whenever we delete the i^{th} row we also delete the i^{th} column and vice versa. The principal minors are denoted as M^{IJ} where I is the set of deleted row indices and J is the set of deleted column indices. Note that I=J for principal minors.

For example, consider the matrix A = \begin{bmatrix} 4 & 9 & 5 & 3 \\ -2 & -3 & 8 & 6 \\ 0 & 0 & -4 & 8 \\ 8 & 1 & 0 & 2 \end{bmatrix}.

Suppose we take I=J=\{2,4\} . This means that we delete the 2^{nd} and 4^{th} rows as well as the 2^{nd} and 4^{th} columns. The principal minor in this case is computed as,

M_{IJ} = \left|\begin{array}{cc}4 & 5\\0 & -4\end{array}\right| = (4)\cdot (-4) - (5)\cdot (0) = -16
##### Leading Principal Minors of a Matrix:

The minors corresponding to the upper left portion of a matrix are known as the leading principal minors of the matrix. Considering the same example as before,

A = \begin{bmatrix} 4 & 9 & 5 & 3 \\ -2 & -3 & 8 & 6 \\ 0 & 0 & -4 & 8 \\ 8 & 1 & 0 & 2 \end{bmatrix}

The leading principal minors are,

det \begin{bmatrix} 4 \end{bmatrix} = 4 det \begin{bmatrix} 4 & 9 \\ -2 & -3 \end{bmatrix} = -30 det \begin{bmatrix} 4 & 9 & 5 \\ -2 & -3 & 8 \\ 0 & 0 & -4 \end{bmatrix} = 120 det \begin{bmatrix} 4 & 9 & 5 & 3 \\ -2 & -3 & 8 & 6 \\ 0 & 0 & -4 & 8 \\ 8 & 1 & 0 & 2 \end {bmatrix} = -5112

If all the leading principal minors of a matrix are positive, the the matrix is a positive definite matrix.

Hey 👋

I'm currently pursuing a Ph.D. in Maths. Prior to this, I completed my master's in Maths & bachelors in Statistics.

I created this website for explaining maths and statistics concepts in the simplest possible manner.

If you've found value from reading my content, feel free to support me in even the smallest way you can.