Minor of a Matrix - Definition, Formula & Example

The Small Determinant Hiding Inside Every Big One

Every determinant of a 3x3 matrix you have ever expanded was secretly built from three smaller 2x2 determinants. Those smaller determinants have a name: they are the minors. Learn to carve a minor out cleanly, and the determinant, the adjoint, and the inverse all become one repeated move instead of three separate mysteries.

What Is A Minor Of A Matrix?

The minor of an element $a_{ij}$ in a square matrix is the determinant of the submatrix that remains after you delete the row and the column that contain that element. It is written $M_{ij}$, matching the row and column of the element it belongs to.

Read the subscripts in order: for $M_{ij}$, delete row $i$ first, then column $j$, and take the determinant of what is left. For a 3x3 matrix, the row index $i$ and the column index $j$ each range over the set ${1, 2, 3}$, so a 3x3 matrix has nine minors - the set ${M_{11}, M_{12}, \dots, M_{33}}$, one for every entry.

Every entry keeps its position in the order of the matrix; the minor just measures the determinant of everything that is not in that entry's row or column.

How Do You Find The Minor Of A Matrix?

The method is three fixed steps, the same for any size:

1. Pick the element $a_{ij}$.

2. Delete row $i$ and column $j$, leaving a smaller square matrix.

3. Take the determinant of that smaller matrix. That number is $M_{ij}$.

For a 2x2 matrix, deleting a row and a column leaves a single entry, so each minor is just that leftover number. Taking

$$A = \begin{bmatrix} a & b \ c & d \end{bmatrix},$$

the minor $M_{11}$ is $d$ (delete row 1 and column 1), and $M_{12}$ is $c$ (delete row 1 and column 2).

For a 3x3 matrix, deleting a row and a column leaves a 2x2 matrix, so each minor is a 2x2 determinant, computed as $(\text{top-left})(\text{bottom-right}) - (\text{top-right})(\text{bottom-left})$.

What Is The Difference Between A Minor And A Cofactor?

This is the distinction people mix up most, and the sibling article on the cofactor matrix approaches it sign-first; here we come at it minor-first. A minor is a plain determinant. A cofactor $C_{ij}$ is that same minor with a position sign attached:

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

When $i + j$ is even the sign is $+$, so the cofactor equals the minor. When $i + j$ is odd the sign is $-$, so the cofactor is the negative of the minor. The signs fall into a fixed checkerboard, top-left always $+$:

$$\begin{bmatrix} + & - & + \ - & + & - \ + & - & + \end{bmatrix}.$$

So a minor measures size; the cofactor adds the sign the position demands. They agree only where the checkerboard shows a $+$.

Where Is The Minor Of A Matrix Used?

Minors are rarely the destination - they are the step that powers three larger results:

• Determinant. Expanding along any row, the determinant is the sum of each entry times its cofactor, and each cofactor is a signed minor. This is cofactor (Laplace) expansion.

• Adjoint. The adjoint is the transpose of the matrix of cofactors, and every cofactor starts life as a minor.

• Inverse. The inverse of a matrix is the adjoint divided by the determinant, valid whenever the determinant is non-zero — so the minors feed the whole pipeline. These links between minors, determinants, and inverses are the heart of matrices and determinants.

Examples Of Minor Of A Matrix

The set runs from a single minor, through the most common deletion mistake, to 2x2 minors, several 3x3 minors, the minor-to-cofactor step, and a full determinant.

Example 1

Find the minor $M_{11}$ of $A = \begin{bmatrix} 4 & 3 & 2 \ 1 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$.

Delete row 1 and column 1, leaving $\begin{bmatrix} 5 & 6 \ 8 & 9 \end{bmatrix}$.

Take its determinant:

$$M_{11} = (5)(9) - (6)(8).$$

$$M_{11} = 45 - 48.$$

$$M_{11} = -3.$$

Final answer: $M_{11} = -3$.

Example 2

Find the minor $M_{23}$ of the same matrix $A$.

Wrong attempt. A student reads $M_{23}$ and deletes row 3 and column 2 — swapping the indices — leaving $\begin{bmatrix} 4 & 2 \ 1 & 6 \end{bmatrix}$ and reporting $22$.

Check the rule. For $M_{ij}$, delete row $i$ then column $j$, so $M_{23}$ deletes row 2 and column 3, not row 3 and column 2. The submatrix above is the wrong one.

Correct. Delete row 2 and column 3, leaving $\begin{bmatrix} 4 & 3 \ 7 & 8 \end{bmatrix}$:

$$M_{23} = (4)(8) - (3)(7).$$

$$M_{23} = 32 - 21.$$

$$M_{23} = 11.$$

Final answer: $M_{23} = 11$.

Example 3

Find the minors $M_{11}$ and $M_{21}$ of $B = \begin{bmatrix} 2 & 7 \ 5 & 3 \end{bmatrix}$.

For a 2x2 matrix each minor is the single leftover entry.

Delete row 1 and column 1: $M_{11} = 3$.

Delete row 2 and column 1: $M_{21} = 7$.

Final answer: $M_{11} = 3$ and $M_{21} = 7$.

Example 4

Find the minor $M_{22}$ of $C = \begin{bmatrix} 1 & 0 & 2 \ 3 & 1 & 0 \ 0 & 2 & 1 \end{bmatrix}$.

Delete row 2 and column 2, leaving $\begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix}$:

$$M_{22} = (1)(1) - (2)(0).$$

$$M_{22} = 1 - 0.$$

$$M_{22} = 1.$$

Final answer: $M_{22} = 1$.

Example 5

Find the cofactor $C_{31}$ of $A$ from Example 1, given its minor.

First the minor. Delete row 3 and column 1, leaving $\begin{bmatrix} 3 & 2 \ 5 & 6 \end{bmatrix}$:

$$M_{31} = (3)(6) - (2)(5) = 18 - 10 = 8.$$

Now apply the sign. Position $(3,1)$ has $i + j = 4$, which is even, so $(-1)^{3+1} = +1$:

$$C_{31} = (+1)(8) = 8.$$

Final answer: $C_{31} = 8$ - here the cofactor equals the minor because the position sign is $+$.

Example 6

Use minors and cofactors to find $\det C$ for $C$ from Example 4 by expanding along the first row.

The determinant is each first-row entry times its cofactor:

$$\det C = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}.$$

Compute the needed cofactors (signed minors). $C_{11} = +\det\begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix} = 1$, and $C_{13} = +\det\begin{bmatrix} 3 & 1 \ 0 & 2 \end{bmatrix} = 6$. The middle term drops out because $a_{12} = 0$:

$$\det C = (1)(1) + (0)(C_{12}) + (2)(6).$$

$$\det C = 1 + 0 + 12.$$

$$\det C = 13.$$

Final answer: $\det C = 13$. The zero entry wiped out a whole term, so choosing a row with zeros saves work.

Common Mistakes

Mistake 1: Deleting the wrong row or column

Where it slips in: Reading the subscripts of $M_{ij}$.

Don't do this: For $M_{23}$, delete row 3 and column 2 by swapping the indices.

The correct way: Delete row $i$ first, then column $j$ — row before column, matching the subscript order. The first instinct when working quickly is to grab whichever row and column look convenient; anchoring on "row then column" every time removes the guesswork.

Mistake 2: Confusing the minor with the cofactor

Where it slips in: Being asked for a cofactor and reporting the minor unchanged.

Don't do this: Write $C_{21}$ equal to $M_{21}$ without checking the position sign.

The correct way: A cofactor is a signed minor: $C_{ij} = (-1)^{i+j} M_{ij}$. Read the checkerboard sign for the position before you commit the value. The minor and cofactor agree only where the sign is $+$.

Mistake 3: Reversing the order in the 2x2 determinant

Where it slips in: Evaluating the leftover 2x2 block.

Don't do this: Compute $bc - ad$ instead of $ad - bc$ for $\begin{bmatrix} a & b \ c & d \end{bmatrix}$.

The correct way: The determinant is main diagonal minus off diagonal: $(a)(d) - (b)(c)$. Reversing the two products flips the sign of every minor that follows.

Conclusion

• The minor of a matrix element $a_{ij}$ is the determinant left after deleting row $i$ and column $j$, written $M_{ij}$.

• Find it in three steps: pick the element, delete its row and column, take the determinant of the rest.

• For a 2x2 matrix each minor is a single entry; for a 3x3 matrix each minor is a 2x2 determinant.

• A cofactor is a signed minor, $C_{ij} = (-1)^{i+j} M_{ij}$; minors and cofactors agree only where the position sign is $+$.

• Minors drive the determinant, the adjoint, and the inverse of a matrix.

To take the minor of a matrix and the rest of the determinants chapter further with a teacher, explore Bhanzu's algebra tutor or high school math tutor, or join structured math classes online.

Practice These To Solidify Your Understanding

1. Find the minor $M_{12}$ of $\begin{bmatrix} 4 & 3 & 2 \ 1 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$.

2. Find the cofactor $C_{12}$ of the same matrix, then compare it with the minor.

3. Find the minor $M_{33}$ of a diagonal matrix $\begin{bmatrix} 2 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 7 \end{bmatrix}$.

If Question 2 comes out equal to Question 1, revisit Mistake 2 — position $(1,2)$ has an odd index sum, so its sign is negative.

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