Determinants and Inverse Matrices

One number decides whether a matrix can be undone.

12 problems · about 22 minutes · TEKS P.3.G

What this lesson teaches

The student computes determinants of 2×2 and 3×3 matrices and finds a 2×2 inverse.

The 2×2 determinant is ad − bc, and it measures area scaling.
A determinant of zero means the matrix is singular and cannot be inverted.
The 2×2 inverse swaps the diagonal, negates the off-diagonal, and divides by the determinant.

Straightforward practice. Get the method working first.

5 problems

[[4, 1], [3, 2]]. What is the determinant?
Answer 5
Why 5.
What does a determinant of zero mean?
Answer The matrix is singular and has no inverse.
Why No inverse exists.
[[5, 2], [3, 4]]. Determinant?
Answer 14
Why 20 − 6.
[[1, 0], [0, 1]]. Determinant?
Answer 1
Why The identity.
[[6, 3], [4, 2]]. Determinant?
Answer 0
Why 12 − 12.

The same ideas with more to keep track of.

3 problems

[[2, 0], [0, 3]]. Determinant?
Answer 6
Why 6 − 0.
A determinant of 0 means an inverse exists? Enter 1 for yes, 0 for no.
Answer 0
Why The formula divides by it.
A transformation has determinant 5. A shape of area 3 becomes what area?
Answer 15
Why Areas scale by the determinant.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the steps for finding a 2×2 inverse in order.
Answer 1. Compute the determinant, ad − bc. 2. Check that it is not zero. 3. Swap a and d, and negate b and c. 4. Divide every entry by the determinant.
Why Check before you divide.
[[7, 3], [2, 1]]. Determinant?
Answer 1
Why 7 − 6.
The Determinant: [[3, 1], [2, 4]]. What is the determinant?
Answer 10
Why 10.
The Singular One: [[2, 4], [1, 2]]. What is the determinant?
Answer 0
Why 0, so the matrix is singular.