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:
Pick the element $a_{ij}$.
Delete row $i$ and column $j$, leaving a smaller square matrix.
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
Find the minor $M_{12}$ of $\begin{bmatrix} 4 & 3 & 2 \ 1 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$.
Find the cofactor $C_{12}$ of the same matrix, then compare it with the minor.
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.
Want a live Bhanzu trainer to walk through more minor-of-a-matrix problems? Book a free demo class — online globally.
Read More
Was this article helpful?
Your feedback helps us write better content