Ever wondered why a 2‑by‑2 grid of numbers can tell you if a system of equations has a unique solution? That's the magic of matrices and determinants!
Think of a matrix as a tidy spreadsheet of numbers. The determinant is a single number that summarises whether that spreadsheet can be ‘unscrambled’ – like checking if a lock has a unique key.
What is a Matrix?
A matrix (plural: matrices) is just a rectangular arrangement of numbers, symbols, or expressions, written in rows and columns. For example, the 2×3 matrix
A = \[\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\end{bmatrix}\]
has 2 rows and 3 columns. The size of a matrix is noted as m × n (read “m by n”).
- Row: a horizontal line of entries.
- Column: a vertical line of entries.
- Element: a single number, written as aij where i = row, j = column.
Why care? In ISC maths, matrices help you solve simultaneous equations, perform linear transformations, and even compute areas in coordinate geometry.
How to Find the Determinant of a Matrix?
The determinant only exists for square matrices (same number of rows and columns). It’s a single value that tells you if the matrix is invertible (i.e., if you can reverse its effect).
Here’s a quick step‑by‑step flowchart:
Step 1 – 2×2 matrix: For \[\begin{bmatrix}a & b\\c & d\end{bmatrix}\] , determinant = ad – bc.
Example: Find det of \[\begin{bmatrix}3 & 5\\2 & 7\end{bmatrix}\]. Compute (3×7) – (5×2) = 21 – 10 = 11.
Step 2 – 3×3 matrix: Use the rule of Sarrus or cofactor expansion. The rule of Sarrus works by adding the products of the three diagonals from left to right and subtracting the products of the three diagonals from right to left.
Example: For \[\begin{bmatrix}1 & 2 & 3\\0 & -4 & 5\\6 & 7 & 8\end{bmatrix}\], apply Sarrus: \[(1·(-4)·8) + (2·5·6) + (3·0·7) - (3·(-4)·6) - (1·5·7) - (2·0·8) = -32 + 60 + 0 - (-72) - 35 - 0 = 65.
Step 3 – Larger matrices (4×4 and beyond): Expand along a row or column that has the most zeros to keep calculations short, or use row‑reduction (Gaussian elimination) to turn the matrix into an upper‑triangular form, then multiply the diagonal entries.
Properties of Determinants (Quick Reference)
| Property | What it means |
|---|---|
| Determinant of a triangular matrix | Just multiply the numbers on the main diagonal. |
| Row swap | Swapping two rows changes the sign (positive ↔ negative). |
| Row multiplication | Multiplying a row by a constant k multiplies the determinant by k. |
| Row addition | Adding a multiple of one row to another leaves the determinant unchanged. |
| Zero row or column | If any row or column is all zeros, determinant = 0. |
Common Mistakes to Avoid
- Trying to find a determinant for a non‑square matrix – it simply doesn’t exist.
- Mixing up the sign when expanding cofactors – remember the checkerboard pattern (+, –, +, – …).
- Forgetting to adjust the sign after a row swap during Gaussian elimination.
- Dropping a minus sign when using the rule of Sarrus – the three “backward” diagonals are subtracted.
Bullet Summary – Your Cheat Sheet
- Matrix: rectangular array of numbers; size = rows × columns.
- Determinant: single number for square matrices; tells if matrix is invertible.
- 2×2: ad – bc.
- 3×3: Sarrus or cofactor expansion.
- 4×4+: expand on a zero‑rich row/column or use row‑reduction.
- Key properties: diagonal product for triangular, sign flips on row swap, scaling rows scales determinant, adding rows doesn’t change it.
- Zero row/column → determinant = 0.
📝 Likely Exam Questions
- Find the determinant of the matrix \[\begin{bmatrix}2 & 3\\5 & 7\end{bmatrix}\].
Answer: (2·7) – (3·5) = 14 – 15 = -1. - Given \[A = \begin{bmatrix}1 & 2 & 0\\-1 & 3 & 4\\0 & 5 & 6\end{bmatrix}\], compute det(A) using cofactor expansion.
Answer: Expand along the first row (contains a zero). det = 1·C11 + 2·C12 = 1·(3·6‑4·5) – 2·((-1)·6‑4·0) = 1·(18‑20) – 2·(-6) = -2 + 12 = 10. - State two properties of determinants that help simplify calculations.
Answer: (i) Swapping two rows changes the sign of the determinant. (ii) Adding a multiple of one row to another leaves the determinant unchanged. - If a 3×3 matrix has a row of zeros, what is its determinant? Explain.
Answer: The determinant is 0 because any expansion will include a factor of 0, indicating the matrix is singular (non‑invertible). - Using Gaussian elimination, transform \[\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}\] to upper‑triangular form and find its determinant.
Answer: Subtract 3·R1 from R2 → \[\begin{bmatrix}1 & 2\\0 & -2\end{bmatrix}\]. Determinant = product of diagonal = 1·(-2) = -2.