The document provides information about calculating inverses of matrices. It defines what an inverse matrix is and that for a matrix A to have an inverse, its determinant must be non-zero. It gives the formula to calculate the determinant of 2x2 matrices and explains how to determine the inverse of a 2x2 matrix using the determinant. Examples are provided to demonstrate calculating the determinant, checking if a matrix is invertible, and calculating the inverse. The document also discusses using the CAS to find the inverse of larger matrices.

2. Learning Intention and Success
Criteria
 Learning Intention: Students will understand the
what types of matrices have inverses, and how the
process for calculating the inverse.
 Success Criteria: You will be able calculate the
determinant of a matrix, explain how the determinant
relates to the inverse, and calculate the inverse of a
matrix

3. Prior Knowledge
 Given a matrix 𝐴 and the identity matrix 𝐼, we have shown
that 𝐴 × 𝐼 = 𝐴 = 𝐼 × 𝐴
Calculate
2 3
3 5
×
5 −3
−3 2
.
2 3
3 5
×
5 −3
−3 2
=
2 × 5 + 3 × −3 2 × −3 + 3 × 2
3 × 5 + 5 × −3 3 × −3 + 5 × 2
=
1 0
0 1

4. What is an inverse?
 If two matrices are multiplied together and the result
is the identity matrix, we say that the matrices are
inverses of one another
 For a matrix 𝐴, the inverse is denoted 𝐴−1.
 The inverse matrix has the property that:
𝐴 × 𝐴−1
= 𝐼 = 𝐴−1
× 𝐴
 Since 𝐴 × 𝐴−1
and 𝐴−1
× 𝐴 are both defined, 𝐴 must
be a square matrix

5. The Determinant
 Determinant: A value for every square matrix which
defines the invertability (whether or not a matrix can
be inverted) of a matrix.
 Notation: det(𝐴) or 𝐴
 For a 2 × 2 matrix, defined as 𝐴 =
𝑎 𝑏
𝑐 𝑑
, the
determinant is calculated as:
det 𝐴 = 𝐴 = 𝑎 × 𝑑 − 𝑏 × 𝑐
 For larger matrices, the determinant can be calculated
using the CAS (Menu  7  3)

6. Examples
Calculate the determinant of the follow matrices
1.
1 2
3 4
2.
5 −2
−7 3
3.
−2 4
3 −6
4.
1 0 −2
3 5 −2
1 5 3

7. Examples
Calculate the determinant of the follow matrices
1.
1 2
3 4
= 1 × 4 − 2 × 3 = −2
2.
5 −2
−7 3
3.
−2 4
3 −6
4.
1 0 −2
3 5 −2
1 5 3

8. Examples
Calculate the determinant of the follow matrices
1.
1 2
3 4
= −2
2.
5 −2
−7 3
= 5 × 3 − −2 × −7 = 1
3.
−2 4
3 −6
4.
1 0 −2
3 5 −2
1 5 3

9. Examples
Calculate the determinant of the follow matrices
1.
1 2
3 4
= −2
2.
5 −2
−7 3
= 1
3.
−2 4
3 −6
= −2 × −6 − 4 × 3 = 0
4.
1 0 −2
3 5 −2
1 5 3

10. Examples
Calculate the determinant of the follow matrices
1.
1 2
3 4
= −2
2.
5 −2
−7 3
= 1
3.
−2 4
3 −6
= 0
4.
1 0 −2
3 5 −2
1 5 3
= 5 (𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝐶𝐴𝑆)

11. What does the determinant mean?
 Given a square matrix 𝐴,
 If det 𝐴 = 0, then 𝐴 does not have an inverse. There is
no matrix that we can multiply 𝐴 by to get the identity
matrix
 Also known as 𝐴 is singular
 If det 𝐴 ≠ 0, then 𝐴 has an inverse. There is a matrix
we can multiply 𝐴 by to get the identity matrix
 Also 𝐴 is known as regular or non-singular

12. Examples
Are each of the matrices below invertable?
1.
1 2
3 4
Yes, non-singular
2.
5 −2
−7 3
Yes, non-singular
3.
−2 4
3 −6
No, singular
4.
1 0 −2
3 5 −2
1 5 3
Yes, non-singular

13. Determining the inverse of a matrix
 To determine the inverse of a 2 × 2 matrix
 Given 𝐴 =
𝑎 𝑏
𝑐 𝑑
𝐴−1
=
1
det 𝐴
×
𝑑 −𝑏
−𝑐 𝑎
=
1
𝑎𝑑 − 𝑏𝑐
×
𝑑 −𝑏
−𝑐 𝑎
=
𝑑
𝑎𝑑 − 𝑏𝑐
−
𝑏
𝑎𝑑 − 𝑏𝑐
−
𝑐
𝑎𝑑 − 𝑏𝑐
𝑎
𝑎𝑑 − 𝑏𝑐

14. Example 1
Show that 𝐹 =
−7 −5
4 3
is invertible, and calculate the
inverse.
det 𝐹 = 𝑎𝑑 − 𝑏𝑐
= −7 3 − −5 4
= −21 + 20
= −1
Since det 𝐹 ≠ 0, 𝐹 is invertible (non-singular)

15. Example 1 Continued
𝐹−1
=
1
𝑎𝑑 − 𝑏𝑐
×
𝑑 −𝑏
−𝑐 𝑎
=
1
−1
×
3 −(−5)
− 4 −7
= −1 ×
3 5
−4 −7
=
−3 −5
4 7

16. Checking Inverses
 To check whether or not two square matrices 𝐴 and 𝐵 are
inverses, multiply them together.
 If 𝐴 × 𝐵 = 𝐼, then they are inverses. Otherwise, they are not.
 Example: Are
1 0 −1
0 2 1
1 3 1
and
2 1.5 −1
−1 −1 1
1 1.5 0
inverses?
1 0 −1
0 2 1
1 3 1
×
2 1.5 −1
−1 −1 1
1 1.5 0
=
1 0 −1
0 1 2
0 0 2
They are not inverses.

17. On the CAS
 To calculate the inverse on the CAS:
 Type the matrix, then l-1. Press Enter.