Bell Work
Decide which are possible and which are not possible. Then figure out
what the order of the resulting matrix is.
7 1 2
A = 1 −1 0
5 0 10
1.
2.
3.
4.
AB
AD
CB
DA
−2 3
B= 4 0
−1 −5
3 −2 8
C=
4 5 1
D= 0
2
3
Identity and Inverse
• Additive Identity
• Additive Inverse
• Multiplicative Identity
• Multiplicative Inverse
Additive Identity for Matrices
The zero matrix is the m x n matrix consisting entirely of zeros.
Multiplicative Identity for Matrices
The identity matrix is the n x n matrix In with 1’s on the main
diagonal (upper left to bottom right) and 0’s everywhere else.
I2 =
I3 =
I4 =
Additive Inverse of Matrices
Let A = [aij] then B = [-aij] is the additive inverse of A because A + B = 0
(the zero matrix).
1
Example: A =
2
4
then the additive inverse of A is B =
3
Multiplicative Inverse of Matrices
Given matrix A, B is A’s inverse if AB = BA = In.
Only square matrices have inverses.
Example
Verify A and B are inverses.
1 2
3 −2
A=
and B =
1 3
−1 1
Determinant of a Matrix
The determinant is a special number associated to any square
matrix.
𝑎
2 x 2 matrix: A =
𝑐
𝑏
, detA = ad – bc.
𝑑
Example
Find the determinant.
3
1
A=
14 −2
Example – You Try
Find the determinant.
1 −4
A=
2 6
Inverse Matrix of a 2x2 Matrix
• If B is the inverse of A we denote the inverse B = A-1.
• An n x n matrix has an inverse iff the determinant ≠ 0.
𝑎
If A =
𝑐
1
𝑏
𝑑
-1
and detA ≠ 0, then A =
𝑑𝑒𝑡𝐴 −𝑐
𝑑
−𝑏
.
𝑎
Example
1 −4
A=
find A-1.
2 6
Example
3 1
A=
find A-1.
4 2
Example – You Try
8 4
A=
find A-1.
10 5
Homework
7.2 (pg. 540-541) #33-36