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Last updated on 13 September 2025
Each element of a square matrix has a minor, which is the determinant of the submatrix formed by removing its row and column. Minors are used to calculate the determinant, cofactors, adjoint, and inverse of a matrix.
The minor of an element in a matrix is the determinant of the submatrix obtained after the removal of its row and column. For an element aij, its minor is denoted by Mij. Let’s consider a matrix B:
\(B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\[0.3em] b_{21} & b_{22} & b_{23} \\[0.3em] b_{31} & b_{32} & b_{33} \end{bmatrix}\)
B12 is in row 1, column 2, so we will remove row 1 and column 2 from the given matrix.
The minor of b12 is:
\(M_{12} = \begin{bmatrix} b_{21} & & b_{23} \\ b_{31} & & b_{33} \end{bmatrix}\)
Similarly, the minor of each element bij in matrix B can be calculated and arranged to form a cofactor matrix M.
\(M = \begin{bmatrix} M_{11} & M_{12} & M_{13} \\[0.3em] M_{21} & M_{22} & M_{23} \\[0.3em] M_{31} & M_{32} & M_{33} \end{bmatrix}\)
We should find the minor of an element inside a matrix. To do that, follow the steps given below:
Step 1: Identify the row and column the element belongs to and remove them from the given matrix.
Step 2: From the remaining elements, form the submatrix and compute its determinant.
Step 3: Find the determinant of the submatrix to get the minor. Using minor values, form a new matrix called matrix of a minor.
For example, consider a 3 × 3 matrix:
\(B = \begin{bmatrix} 1 & 2 & 3 \\[0.3em] 0 & 4 & 5 \\[0.3em] 7 & 6 & 8 \end{bmatrix}\)
To find the minor of element b11 remove the first row and first column of the original matrix.
Minor of b11 = M11 = \( \Bigg | \begin{matrix} 4 & 5\\ 6 & 8 \end{matrix} \Bigg|\)
= (4 × 8) – (5 × 6) = 32 – 30 = 2
The minor of element b23 can be found by removing the second row and third column.
Minor of b23 = M23 = \( \Bigg | \begin{matrix} 1 & 2\\ 7 & 6 \end{matrix} \Bigg|\)
= (1 × 6) – (2 × 7) = 6 – 14 = – 8
To find the minor of element b32 remove the third row and second column.
Minor of b32 = M32 = \( \Bigg | \begin{matrix} 1 & 3\\ 0 & 5 \end{matrix} \Bigg|\)
= (1 × 5) – (3 × 0) = 5 – 0 = 5
We can find the minor of each element in this manner. By finding minors of elements, we can form a new matrix that will be the minor of matrix B.
\(M = \begin{bmatrix} M_{11} & M_{12} & M_{13} \\[0.3em] M_{21} & M_{22} & M_{23} \\[0.3em] M_{31} & M_{32} & M_{33} \end{bmatrix} = \begin{bmatrix} 2 & -35 & -28 \\[0.3em] 18 & 9 & -9 \\[0.3em] 11 & 5 & 8 \end{bmatrix}\)
A matrix’s cofactor, determinant, adjoint, and inverse can be determined by finding the minor of a matrix. Let us see the applications of the minor of a matrix.
Cofactor Matrix: Cij is used to denote the cofactor of an element in a matrix. Cij is determined by multiplying the minor Mij and (–1)i + j.
Therefore, Cij = (–1)i + j Mij
\(\text {Cofactor Matrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\[0.3em] C_{21} & C_{22} & C_{23} \\[0.3em] C_{31} & C_{32} & C_{33} \end{bmatrix}\)
We get cofactor matrix when we replace each element with its cofactor.
Determinant of a matrix: The determinant of a matrix is a single value that summarizes properties of a matrix. It is calculated using cofactor expansion. The determinant can be found by following these steps:
In a matrix D, the determinant is denoted by |d| or det D.
The determinant formula for the elements of the first row of matrix D will be:
|d| = d11C11 + d12C12 + d13C13
Here, Cij = (–1)i+j Mij. So, the determinant of matrix D is:
\(|D| = {{d_{11}(-1)^{1+1} \space \bigg| {\begin{matrix} d_{22} & & d_{23} \\ d_{32} & & b_{33} \end{matrix} } \bigg |} \space + \space {d_{12}(-1)^{1+2}\space \bigg| {\begin{matrix} d_{21} & & d_{23} \\ d_{31} & & b_{33} \end{matrix} } \bigg |} \space + \space {d_{13}(-1)^{1+3} \space \bigg| {\begin{matrix} d_{21} & & d_{21} \\ d_{31} & & b_{32} \end{matrix} } \bigg |}}\)
Adjoint of a matrix: To find the adjoint of a 3 × 3 matrix:
Let us consider matrix B:
\(B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\[0.3em] b_{21} & b_{22} & b_{23} \\[0.3em] b_{31} & b_{32} & b_{33} \end{bmatrix}\)
\(\text {Cofactor (B)} = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\[0.3em] C_{21} & C_{22} & C_{23} \\[0.3em] C_{31} & C_{32} & C_{33} \end{bmatrix}\)
The adjoint of matrix B is equal to the transpose of cofactor matrix
i.e., adj(B) = transpose of cofactor matrix of B
\(adj(B) = [\text {Cofactor (B)}]^T = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\[0.3em] C_{12} & C_{22} & C_{32} \\[0.3em] C_{13} & C_{23} & C_{33} \end{bmatrix}\)
Inverse of a matrix: We should divide the adjoint matrix by the determinant to find the inverse of a matrix.
For a matrix D:
\(D = \begin{bmatrix} d_{11} & d_{12} & d_{13} \\[0.3em] d_{21} & d_{22} & d_{23} \\[0.3em] d_{31} & d_{32} & d_{33} \end{bmatrix}\)
The inverse is \(D⁻¹ = {{1 \over det (D)} \times adj(D)}\)
Determinant:
|d| = d11C11 + d12C12 + d13C13
Where, Cij is the cofactor of element dij .
Adjoint:
\(adj(D) = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\[0.3em] C_{12} & C_{22} & C_{32} \\[0.3em] C_{13} & C_{23} & C_{33} \end{bmatrix}\)
Using inverse formula, If \(\text {det D} \neq 0\) the inverse of D is:
\(D⁻¹ = {{1 \over det (D)} \times adj(D)}\)
A minor is a building block for the calculation of determinants, cofactors, and matrix inverses. It can be applied to the following real-life applications:
It is possible for students to make avoidable errors while solving for the minor of a matrix. Understanding these commonly occurring errors can help avoid incorrect results.
Find the minor a11 = 4
M11 = 3
Given Matrix:
\(A = \begin{bmatrix} 4 & 5 \\ 2 & 3 \end{bmatrix}\)
Delete row 1 and column 1
Only a22 remains.
\(M_{11} = \begin{bmatrix} a_{22} \end{bmatrix} = 3\)
Find the minor of a12 = 2
M12 = – 6
Given matrix:
\(A = \begin{bmatrix} 1 & 2 & 3 \\[0.3em] 4 & 5 & 6 \\[0.3em] 7 & 8 & 9 \end{bmatrix}\)
Delete row 1 and column 2
\(M_{12} = \begin{bmatrix} 4 & 6 \\ 7 & 9 \end{bmatrix}\)
M12 = (4 × 9) – (6 × 7) = 36 – 42 = – 6
Find the minor of b22
M22 = – 10
Given matrix:
\(B = \begin{bmatrix} 2 & 1 & 4 \\[0.3em] 0 & 3 & 5 \\[0.3em] 6 & 2 & 7 \end{bmatrix}\)
Delete row 2, column 2:
\(M_{22} = \begin{bmatrix} 2 & 4 \\ 6 & 7 \end{bmatrix}\)
M22 = (2 × 7) – (4 × 6) = 14 – 24 = – 10
Find the minor of C13 = 2
M13 = – 20
Given matrix:
\(C = \begin{bmatrix} 3 & -1 & 2 \\[0.3em] 0 & 5 & 1 \\[0.3em] 4 & 6 & -2 \end{bmatrix}\)
Delete row 1 and column 3
\(M_{13} = \begin{bmatrix} 0 & 5 \\ 4 & 6 \end{bmatrix}\)
M13 = (0 × 6) – (5 × 4) = 0 – 20 = – 20
Find the minor of d21, = 4
M21 = – 65
Given matrix:
\(D_{21} = \begin{bmatrix} 1&0 & 2 & 3 \\ 4&5&6&1 \\ 7&8 & 9 & 0 \\ 2&3&4&5 \end{bmatrix}\)
Delete row 2 and column 1:
\(M_{21} = \begin{bmatrix} 0 & 2 & 3 \\ 8 & 9 & 0 \\ 3&4&5 \end{bmatrix}\)
M21 = {0 × (9 × 5 – 0 × 4)} – {2 × (8 × 5 – 0 × 3)} + {3 × (8 × 4 – 9 × 3)}
= 0 – {2 × (40)} + {3 × (32 – 27)}
= 0 – 80 + {3 × (5)}
= – 80 + 15
= – 65
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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