Matrix Algebra Assignment
Let
A=
4
−1
6
9
and
B=
0
3
3
−2
Find:
1. A + B
2. AB
3. BA
4. A' (the transpose of A).
5. What is an Identity Matrix?
To solve the given questions for the matrices A and B, we will perform basic operations in matrix algebra. Here's how you can find the solutions:
1. A + B:
To find the sum of two matrices, simply add the corresponding elements from each matrix. The addition operation is performed element-wise.
A + B =
(4 + 0) (-1 + 3)
(6 + 3) (9 + (-2))
Therefore, A + B is:
4 2
9 7
2. AB:
To find the matrix product AB, we need to multiply the elements of the rows of matrix A with the corresponding elements of the columns of matrix B.
AB =
(4 * 0 + (-1) * 3) (4 * 3 + (-1) * (-2))
(6 * 0 + 9 * 3) (6 * 3 + 9 * (-2))
Therefore, AB is:
-3 14
27 -3
3. BA:
To find the matrix product BA, we perform the same multiplication process as in step 2.
BA =
(0 * 4 + 3 * 6) (0 * (-1) + 3 * 9)
(3 * 4 + (-2) * 6) (3 * (-1) + (-2) * 9)
Therefore, BA is:
18 27
6 -21
4. A':
To find the transpose of matrix A, simply swap the elements across the main diagonal of the matrix. That is, each element A_ij in A becomes A_ji (where i is the row and j is the column).
4 6
-1 9
Therefore, A' (A transpose) is:
4 -1
6 9
5. Identity Matrix:
An identity matrix is a square matrix where the elements on the main diagonal (from the top left to the bottom right) are equal to 1, while all the other elements are zero. In simpler terms, it is like the "identity" of matrices because when you multiply any matrix by the identity matrix, it remains unchanged.
For example, the 2x2 identity matrix I_2 is:
1 0
0 1
In general, the identity matrix I_n (for an n x n matrix) is:
1 0 0 ... 0
0 1 0 ... 0
0 0 1 ... 0
.....................
0 0 0 ... 1
I hope this helps you understand the steps to solve the given matrix algebra questions.