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Determinants and inverses
The determinant gives the area scale factor of a matrix transformation and determines whether the matrix is invertible. A non-zero determinant permits an inverse, and this lesson finds it in the 2 × 2 and 3 × 3 cases.
Builds on Matrix algebra and transformations.
IN THIS TOPIC
- Evaluate 2 × 2 and 3 × 3 determinants and interpret them as scale factors.
- Recognise singular matrices and describe what a zero determinant does to space.
- Find and verify inverses of 2 × 2 and 3 × 3 matrices, and use the reversal rule for products.
COMMON MISCONCEPTION
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|, and a negative sign flags that orientation has flipped.
A 3 × 3 determinant expands along its top row. Each entry multiplies the 2 × 2 determinant left after deleting its own 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. In three dimensions that number is the volume scale factor, so the unit cube becomes a parallelepiped of volume 13.
When the determinant is zero
When the determinant is zero the matrix is singular, and the geometry explains the algebra. Zero scale factor means the whole plane is flattened onto a line, or onto the origin. Distinct points then share images, so no transformation can reverse the mapping and no inverse exists. There is an entire family of such matrices, one for every zero determinant.
The 2 × 2 inverse
For det ≠ 0 the inverse exists and undoes the matrix, so A⁻¹A = AA⁻¹ = I. In the 2 × 2 case there is 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 by multiplying. MM⁻¹ comes out as rows (1, 0), (0, 1), the identity. That check is one multiplication and it 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.
GUIDED PRACTICE
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 3 × 3 inverse
Three by three needs a longer route and it is worth drilling, because the arithmetic is where the marks go, not the idea. Compute the determinant. Replace every entry by its minor, the 2 × 2 determinant left when you delete that entry's row and column. Apply the chequerboard of signs to turn minors into cofactors. Transpose. Divide by the determinant.
WORKED EXAMPLE
Inverting a 3 × 3 matrix
Find the inverse of M with rows (1, 2, 3), (0, 1, 4), (5, 6, 0).
Determinant along the top row: 1(0 − 24) − 2(0 − 20) + 3(0 − 5) = −24 + 40 − 15 = 1.
Matrix of minors, rows (−24, −20, −5), (−18, −15, −4), (5, 4, 1).
Apply the signs to get cofactors, rows (−24, 20, −5), (18, −15, 4), (5, −4, 1), then transpose.
Since det = 1, the inverse is rows (−24, 18, 5), (20, −15, −4), (−5, 4, 1). Multiply back to I before writing anything else down.
In practice you will meet the 3 × 3 inverse mainly through simultaneous equations, and a calculator can produce it in an instant. The 'show that' versions still want the minors and the transpose visible, so keep the method at your fingertips even when a calculator could produce the answer.
ASSESSMENT FOCUS
- 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.
- Do not forget the transpose in the 3 × 3 inverse. Cofactors alone are the commonest lost mark on this topic.
- Verify one inverse per paper by multiplying back to I. It takes one multiplication.
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
12 − 2k = 0, so k = 6. With rows (4, 6), (2, 3), both columns lie along the direction (2, 1), so the whole plane is flattened onto the line y = x/2 and no inverse exists.
det = ad − bc gives the area scale factor as its size, with the sign recording orientation, and a 3 × 3 determinant does the same for volume.
Zero determinant means the plane is flattened onto a line or a point, so there is no inverse.
2 × 2 inverse: swap the diagonal, negate the off-diagonal, divide by the determinant.
3 × 3 inverse: minors, signs, transpose, divide.
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.
Or read them with their worked answers on the determinants and inverses questions page.
CHECK YOUR PROGRESS
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- Evaluate 2 × 2 and 3 × 3 determinants and interpret them as scale factors.
- Recognise singular matrices and describe what a zero determinant does to space.
- Find and verify inverses of 2 × 2 and 3 × 3 matrices, and use the reversal rule for products.
Open the full revision checklist to see every objective in the course in one place.