Maths › Matrices › Determinants and inverses
Determinants and inverses
One number that says what a matrix does to area, whether it can be undone, and how: the determinant, and the inverse that follows from it.
Builds on Matrix algebra and transformations.
IN THIS TOPIC
- Evaluate 2 × 2 and 3 × 3 determinants, and interpret the 2 × 2 case as an area scale factor.
- Recognise singular matrices and what a zero determinant does to the plane.
- Find and verify inverses, and use the reversal rule for products.
WHAT YOU PROBABLY THINK
Every matrix has an inverse, just as every non-zero number has a reciprocal.
The determinant
For a 2 × 2 matrix with rows (a, b) and (c, d), the determinant is ad − bc, and it measures the transformation's effect on area: the unit square maps to a parallelogram of area |ad − bc|, with a negative sign flagging that orientation has flipped.
A 3 × 3 determinant expands along its top row: each entry multiplies the 2 × 2 determinant left by deleting its row and column, with signs +, −, + across the row. For rows (1, 2, 0), (0, 1, 3), (2, 0, 1) the expansion gives 1(1 − 0) − 2(0 − 6) + 0 = 13.
Singular means squashed
When the determinant is zero the matrix is singular, and the geometry explains the algebra: zero area scale factor means the whole plane is flattened onto a single line (or the origin). Distinct points share images, so no transformation can undo the damage, and no inverse exists. That is the opening claim's mistake: matrices have a whole family of non-invertible members, one for every zero determinant.
The inverse
For det ≠ 0 the inverse exists and undoes the matrix: A⁻¹A = AA⁻¹ = I. In the 2 × 2 case it has a closed recipe: swap the diagonal entries, negate the other two, divide by the determinant.
WORKED EXAMPLE
An inverse, found and checked
Find the inverse of the matrix M with rows (3, 1), (1, 2).
det M = 3 × 2 − 1 × 1 = 5.
Swap, negate, divide: M⁻¹ = (1/5) × the matrix with rows (2, −1), (−1, 3).
Check: MM⁻¹ multiplies out to rows (1, 0), (0, 1), the identity. The check is one multiplication and catches sign slips immediately.
For products the inverse reverses the order: (AB)⁻¹ = B⁻¹A⁻¹. Undoing 'B then A' means undoing A first, the same logic as unwrapping anything done in layers.
YOUR TURN
Two determinants and an inverse
Find the determinants of the matrices with rows (5, 2), (7, 3) and rows (6, 4), (3, 2); invert whichever is invertible.
Show the working
First: det = 15 − 14 = 1. Second: det = 12 − 12 = 0, singular, no inverse.
The first inverts: with determinant 1, the inverse is just swap and negate, rows (3, −2), (−7, 5).
A quick multiplication confirms the product is I.
THE EXAM BIT
- State the determinant before writing any inverse; it justifies existence and supplies the divisor.
- Keep the 1/det outside the matrix until the final line to avoid fraction clutter.
- For 3 × 3 expansions, write the +, −, + signs above the top row before starting.
- Verify one inverse per paper by multiplying back to I; it is cheap insurance.
CHECK YOURSELF
The matrix with rows (4, k), (2, 3) is singular. Find k, and describe what the transformation does to the plane at that value.
Show a hint
Set the determinant to zero.
Show the answer
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det = ad − bc is the area scale factor; zero determinant means squashed flat, no inverse.
2 × 2 inverse: swap the diagonal, negate the off-diagonal, divide by the determinant.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Evaluate 2 × 2 and 3 × 3 determinants, and interpret the 2 × 2 case as an area scale factor.
- Recognise singular matrices and what a zero determinant does to the plane.
- Find and verify inverses, and use the reversal rule for products.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.