What Are Minors And Cofactors?
Minors and cofactors are two related numbers attached to each entry of a square matrix. The minor $M_{ij}$ of the entry in row $i$, column $j$ is the determinant of the smaller matrix you get after deleting row $i$ and column $j$. The cofactor $C_{ij}$ is that minor with a plus or minus sign in front, decided by the entry's position.
The link between them is one short formula:
$$C_{ij} = (-1)^{i+j} , M_{ij}$$
Both are defined only for a square matrix, because only square matrices have a determinant. A minor is always a single number, never a leftover grid: you compute the determinant of the smaller grid, and that number is the minor.
Here is the running matrix used throughout this article:
$$A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 1 & 0 & 6 \end{bmatrix}$$
Its order is 3 × 3, and every entry is named $a_{ij}$, where $i$ is the row and $j$ is the column. So $a_{23} = 5$ sits in row 2, column 3. The rest of the article finds the minors and cofactors of this one matrix, then uses them to build its determinant, adjoint, and inverse.
How Do You Find The Minor Of A Matrix?
To find the minor $M_{ij}$, delete row $i$ and column $j$, then take the determinant of the smaller matrix that remains. For a 3 × 3 matrix, that smaller matrix is 2 × 2, so each minor is a simple $ad - bc$ calculation.
Take the top-left entry $a_{11} = 1$. Delete row 1 and column 1:
$$M_{11} = \begin{vmatrix} 4 & 5 \ 0 & 6 \end{vmatrix} = (4)(6) - (5)(0) = 24$$
Now the entry $a_{12} = 2$. Delete row 1 and column 2:
$$M_{12} = \begin{vmatrix} 0 & 5 \ 1 & 6 \end{vmatrix} = (0)(6) - (5)(1) = -5$$
Working through every position of $A$ gives all nine minors:
Table: The nine minors of matrix $A$.
Position | Delete | Minor $M_{ij}$ |
|---|---|---|
$M_{11}$ | row 1, col 1 | $24$ |
$M_{12}$ | row 1, col 2 | $-5$ |
$M_{13}$ | row 1, col 3 | $-4$ |
$M_{21}$ | row 2, col 1 | $12$ |
$M_{22}$ | row 2, col 2 | $3$ |
$M_{23}$ | row 2, col 3 | $-2$ |
$M_{31}$ | row 3, col 1 | $-2$ |
$M_{32}$ | row 3, col 2 | $5$ |
$M_{33}$ | row 3, col 3 | $4$ |
For more practice isolating the smaller grid before you evaluate it, see minor of a matrix.
What Is The Cofactor Of A Matrix?
The cofactor $C_{ij}$ is the minor $M_{ij}$ with a sign chosen by its position: $C_{ij} = (-1)^{i+j} M_{ij}$. When $i + j$ is even the cofactor equals the minor; when $i + j$ is odd the cofactor is the minor with its sign flipped.
You rarely need to compute $(-1)^{i+j}$ by hand. The signs follow a fixed checkerboard, the sign chart:
$$\begin{bmatrix} + & - & + \ - & + & - \ + & - & + \end{bmatrix}$$
Read it like a chessboard. The top-left is always $+$, and the sign alternates across every row and down every column. Apply the chart to the nine minors of $A$:
Table: From minors to cofactors, using the sign chart.
Position | Sign | Minor | Cofactor $C_{ij}$ |
|---|---|---|---|
$C_{11}$ | $+$ | $24$ | $24$ |
$C_{12}$ | $-$ | $-5$ | $5$ |
$C_{13}$ | $+$ | $-4$ | $-4$ |
$C_{21}$ | $-$ | $12$ | $-12$ |
$C_{22}$ | $+$ | $3$ | $3$ |
$C_{23}$ | $-$ | $-2$ | $2$ |
$C_{31}$ | $+$ | $-2$ | $-2$ |
$C_{32}$ | $-$ | $5$ | $-5$ |
$C_{33}$ | $+$ | $4$ | $4$ |
Collect the cofactors into a grid and you have the cofactor matrix of $A$:
$$\text{cofactor matrix} = \begin{bmatrix} 24 & 5 & -4 \ -12 & 3 & 2 \ -2 & -5 & 4 \end{bmatrix}$$
The one thing to hold on to: the minor is a raw determinant, and the cofactor is that determinant after the sign chart has spoken.
How Do Minors And Cofactors Give The Determinant?
The determinant of a square matrix is the sum of the entries of any one row (or column), each multiplied by its own cofactor. This is called cofactor expansion, and it turns a 3 × 3 determinant into three 2 × 2 ones.
Expanding $A$ along row 1:
$$\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$
Substitute the row-1 entries $(1, 2, 3)$ and their cofactors $(24, 5, -4)$:
$$\det(A) = (1)(24) + (2)(5) + (3)(-4)$$
$$\det(A) = 24 + 10 - 12 = 22$$
Final answer: $\det(A) = 22$.
You can expand along any row or column and get the same number, which is a useful self-check. Choosing the row or column with the most zeros makes the arithmetic shortest. For the wider picture of how determinants behave, see determinants and matrices and determinants.
How Do Cofactors Build The Adjoint And Inverse?
The adjoint is the transpose of the cofactor matrix, and the inverse is the adjoint divided by the determinant. This is the main payoff: the cofactors you already found do all the work.
First the adjoint, written $\text{adj}(A)$. Transpose the cofactor matrix, so rows become columns:
$$\text{adj}(A) = \begin{bmatrix} 24 & -12 & -2 \ 5 & 3 & -5 \ -4 & 2 & 4 \end{bmatrix}$$
Then the inverse uses the formula $A^{-1} = \dfrac{1}{\det(A)},\text{adj}(A)$. Since $\det(A) = 22 \neq 0$, the inverse exists:
$$A^{-1} = \frac{1}{22} \begin{bmatrix} 24 & -12 & -2 \ 5 & 3 & -5 \ -4 & 2 & 4 \end{bmatrix}$$
If the determinant had come out as $0$, the matrix would be singular and no inverse would exist, no matter how neat the cofactors looked. The full step-by-step method lives at inverse of a 3×3 matrix.
Why Does The Cofactor Sign Chart Exist?
The alternating signs are not decoration, and they are not a rule to memorise for its own sake. They exist so that cofactor expansion agrees with the determinant no matter which row or column you expand along.
The determinant must be one fixed number. A 3 × 3 matrix has three rows and three columns, so there are six different ways to expand it. Every one of them has to give the identical answer. The only sign pattern that keeps all six in agreement is the checkerboard $(-1)^{i+j}$.
It records an orientation, not an accident. Swapping two rows of a matrix flips the sign of its determinant. The alternating sign in the cofactor is the same flip, tracked one entry at a time. It is the bookkeeping that remembers how many swaps a position sits away from the main diagonal.
That is the deep reason the top-left corner is always $+$ and the sign alternates outward. The pattern is forced by consistency, not chosen by convention. Once a student sees that the sign chart is the price of a well-defined determinant, the $(-1)^{i+j}$ stops looking arbitrary.
Who Discovered Cofactors And Determinants?
The idea of expanding a large array into smaller signed pieces is older than the word "determinant," and it appeared on opposite sides of the world within a decade.
Two later mathematicians shaped the version students learn today:
Pierre-Simon Laplace (1749–1827, France) formalised cofactor expansion in 1772, which is why expanding a determinant by minors and cofactors is also called the Laplace expansion.
Augustin-Louis Cauchy (1789–1857, France) fixed the modern meaning of the word "determinant" in 1812 and organised the sign rules into the consistent theory we still use.
Where Are Minors And Cofactors Used In The Real World?
The same expand-into-signed-pieces idea runs under more technology than its textbook page suggests.
Solving systems of equations: Cramer's rule solves for each unknown as a ratio of determinants, and every one of those determinants is built from cofactors.
Computer graphics and robotics: inverting a transformation matrix (to undo a rotation or a camera move) runs on the adjoint, which is nothing but the transposed cofactor matrix.
Cryptography: ciphers that encrypt with a key matrix decrypt with its inverse, and that inverse is assembled from cofactors and the determinant.
Engineering and physics: stress analysis, electrical networks, and control systems all reach a point where a matrix must be inverted, and cofactors are one standard route.
Economics: input-output models of an economy are solved by inverting a large coefficient matrix, again through the determinant-and-adjoint path.
One small idea, deleting a row and a column and tracking a sign, quietly supports code-breaking, animation, and national accounts alike. Mathematics keeps reusing its best tools across fields that look unrelated.
What Are The Most Common Minors And Cofactors Mistakes?
These four errors account for most lost marks on this topic, verified against a UiTM "common errors in mathematics" reference and the sign-rule notes on the Cuemath and GeeksforGeeks minor-and-cofactor pages.
Treating the minor as the cofactor.
Where it slips in:
A student computes the minor $M_{ij}$, writes it straight into the cofactor matrix, and never applies the sign chart.
Don't do this:
Do not skip the $(-1)^{i+j}$ step. The minor and the cofactor are equal only when $i + j$ is even.
The correct way:
After every minor, look up the position on the sign chart. For $C_{12}$ the sign is minus, so $M_{12} = -5$ becomes $C_{12} = +5$.
Deleting the wrong row or column.
Where it slips in:
While finding $M_{23}$, a student deletes row 3 and column 2 instead of row 2 and column 3, then evaluates the wrong 2 × 2 grid.
Don't do this:
Do not read the indices out of order. In $M_{ij}$ the first index is the row, the second is the column.
The correct way:
Point at the entry first, then cover the row it lives in and the column it lives in. Only the four corners that remain go into the minor.
Confusing the small matrix with its determinant.
Where it slips in:
A student writes the leftover 2 × 2 grid and calls that the minor, leaving the answer as a matrix rather than a number.
Don't do this:
Do not stop at the smaller grid. A minor is a single number.
The correct way:
Always finish by taking the determinant of the smaller grid: for a 2 × 2 that is $ad - bc$, and that value is the minor.
Trying to find minors of a non-square matrix.
Where it slips in:
A student is handed a 2 × 3 matrix and starts deleting rows and columns to build cofactors.
Don't do this:
Do not compute minors or cofactors for a rectangular matrix. They are defined only where a determinant exists.
The correct way:
Check that the matrix is square first. If the number of rows does not equal the number of columns, minors and cofactors do not apply.
Practice Problems On Minors And Cofactors
Use $P = \begin{bmatrix} 2 & 3 & 1 \ 1 & 0 & 2 \ 4 & 1 & 3 \end{bmatrix}$ for every problem. Answers follow each line.
Find the minor $M_{11}$ of $P$.
(Answer: delete row 1 and column 1, $\begin{vmatrix} 0 & 2 \ 1 & 3 \end{vmatrix} = -2$.)Find the cofactor $C_{12}$ of $P$.
(Answer: $M_{12} = \begin{vmatrix} 1 & 2 \ 4 & 3 \end{vmatrix} = -5$; sign is minus, so $C_{12} = 5$.)What sign does the chart give position $(2,3)$?
(Answer: $(-1)^{2+3} = -1$, a minus.)Find the cofactor $C_{23}$ of $P$.
(Answer: $M_{23} = \begin{vmatrix} 2 & 3 \ 4 & 1 \end{vmatrix} = -10$, so $C_{23} = -(-10) = 10$.)Compute $\det(P)$ by expanding along row 1.
(Answer: $(2)(-2) + (3)(5) + (1)(1) = -4 + 15 + 1 = 12$.)Are the minor and cofactor of $a_{11}$ equal here?
(Answer: yes, because $1 + 1 = 2$ is even, so $C_{11} = M_{11}$.)
Where Should You Go Next After Minors And Cofactors?
Minors and cofactors are the gateway to every determinant-based technique, and a few natural doors open from here.
Determinant of a matrix. See cofactor expansion applied to larger matrices and compared with other ways of computing a determinant.
Inverse of a matrix. Follow the adjoint-over-determinant route all the way to a checked inverse, including when it fails.
Cramer's rule. Use determinants built from cofactors to solve a whole system of equations in one shot.
If your child is building these foundations, a live Bhanzu trainer teaches minors and cofactors starting from the "why" (the consistency that forces the sign chart, and the inverse the cofactors were built to reach) in the Bhanzu algebra program.
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