12.1 Practice Matrix Operations Answers
Matrix Operations
In this section, we will practice performing various operations on matrices. Matrices are a fundamental concept in linear algebra and are used to represent systems of equations.
1. Matrix Addition
Problem: Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. Find A + B.
Answer:
A + B =
| 1+5 | 2+6 |
|---|---|
| 3+7 | 4+8 |
= [[6, 8], [10, 12]]
2. Matrix Subtraction
Problem: Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. Find A - B.
Answer:
A - B =
| 1-5 | 2-6 |
|---|---|
| 3-7 | 4-8 |
= [[-4, -4], [-4, -4]]
3. Scalar Multiplication
Problem: Let A = [[1, 2], [3, 4]] and k = 2. Find kA.
Answer:
kA =
| 2*1 | 2*2 |
|---|---|
| 2*3 | 2*4 |
= [[2, 4], [6, 8]]
4. Matrix Multiplication
Problem: Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. Find AB.
Answer:
AB =
| 15 + 27 | 16 + 28 |
|---|---|
| 35 + 47 | 36 + 48 |
= [[19, 22], [43, 50]]
5. Matrix Transpose
Problem: Let A = [[1, 2], [3, 4]]. Find A^T.
Answer:
A^T =
| 1 | 3 |
|---|---|
| 2 | 4 |
= [[1, 3], [2, 4]]
Conclusion
In this practice, we have seen how to perform various operations on matrices, including addition, subtraction, scalar multiplication, matrix multiplication, and finding the transpose of a matrix. Mastering these operations is essential for working with matrices and solving systems of linear equations.
