Matrices and Determinants
Determinants, Minors and Cofactors
A single number that decides whether a matrix is invertible, the 2 × 2 rule, and expansion by minors and cofactors for larger matrices.
What this page covers
- Compute a 2 × 2 determinant
- Find the minor of an entry
- Attach the correct cofactor sign
- Expand a 3 × 3 determinant along any row or column
The 2×2 determinant
For A = [[a, b], [c, d]]:
Multiply down the main diagonal, multiply down the other diagonal, subtract.
The source's example
For [[7, 2], [3, -1]]:
Both products carry their own signs. Here (7)(-1) = -7 and the subtraction of 6 makes it -13, not -1. Writing each product in brackets before combining prevents the slip.
The determinant already appeared in the inverse formula, where it is the quantity divided by. That is its primary role: a zero determinant means the division is impossible and no inverse exists.
Minors
The minor Mij of the entry aij is the determinant of the matrix formed by deleting the ith row and the jth column.
The source's own example is the minor of the central entry of a 3 × 3 matrix. Deleting row 2 and column 2 leaves the four corner entries:
Each minor of a 3 × 3 matrix is a 2 × 2 determinant, which is why expansion reduces a larger problem to several smaller ones.
Cofactors
The factor (-1)i+j is +1 when i + j is even and -1 when it is odd. The result is a checkerboard:
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | + | - | + |
| Row 2 | - | + | - |
| Row 3 | + | - | + |
Start with + in the top-left and alternate in both directions. That is faster and more reliable than computing (-1)i+j each time, and the source writes the powers explicitly as (-1)2+1, (-1)2+2, (-1)2+3 when expanding along the second row.
Expansion
Choose any row or column. Multiply each entry by its cofactor and add. The result is the determinant, and it is the same whichever row or column is chosen.
Worked example — expanding along the first row
For [[5, 2, -2], [3, 4, 0], [-4, 2, 6]], the source expands along row 1.
The same matrix, expanded along the first column
The source repeats the calculation down column 1 as a check.
Two independent expansions must agree. They use different entries and different minors, so agreement is strong evidence that both are right.
In the matrix above the (2,3) entry is zero, so expanding along row 2 or column 3 removes one term entirely. A row of two zeros reduces a 3 × 3 expansion to a single 2 × 2 determinant.
What the determinant decides
| |A| | Inverse | System Ax = b | Name |
|---|---|---|---|
| Non-zero | Exists and is unique | Exactly one solution | Non-singular, invertible |
| Zero | Does not exist | No solution, or infinitely many | Singular |
The source records the practical statement as |A| = ad - bc, and if |A| = 0, no solution. Strictly the zero case means no unique solution: the system may still be dependent with infinitely many, which is what the row-reduction methods distinguish.
The inverse formula shows why. A-1 has 1ad - bc as a factor, so a zero determinant makes the whole expression undefined. The source writes the 2 × 2 inverse in cofactor form as
Common mistakes
| Mistake | Correct | Why |
|---|---|---|
| |A| = ad + bc | ad - bc | The diagonals subtract |
| Losing a sign in (7)(-1) - (2)(3) | = -13 | Bracket each product |
| Omitting the cofactor signs | The checkerboard is essential | The pattern alternates from + at the top left |
| Deleting the wrong row or column for a minor | Delete the row and column of the entry | M22 deletes row 2 and column 2 |
| Expanding along a row with no zeros | Choose the row with most zeros | Each zero removes a term |
| Taking a determinant of a non-square matrix | Only square matrices have one | The definition requires it |
Frequently asked questions
What does the determinant tell you?
Whether the matrix is invertible. A non-zero determinant means an inverse exists and the corresponding system has exactly one solution; zero means no inverse and either no solution or infinitely many.
What is a minor?
The determinant of the matrix left after deleting the row and column containing the chosen entry. The source defines it exactly this way.
Where does the cofactor sign come from?
From (-1)i+j, which produces a checkerboard pattern starting with + in the top-left corner. It is easier to picture the pattern than to compute the power each time.
Which row should I expand along?
Whichever has the most zeros. Every zero entry kills a whole minor, so a row with two zeros reduces a 3 × 3 expansion to one term.
Source. Handwritten teaching notes, Week 12, pages 11-14.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.