matrices: A=[-5 4 -6 B=[-2 4 -5
9 -8 7 8 9 3
4 -3 9] 5 -2 2]
AB=?
please show the work if you can.
To find the product of matrices A and B, we need to multiply each element of A with the corresponding element in B and add them up. Here's how to do it:
1. Let's label the elements of matrix A as a_11, a_12, a_13, a_21, a_22, a_23, a_31, a_32, and a_33. Similarly, label the elements of matrix B as b_11, b_12, b_13, b_21, b_22, b_23, b_31, b_32, and b_33.
2. Now, we can calculate the elements of the resulting matrix AB.
- The element in the first row and first column of AB is given by:
AB_11 = a_11 * b_11 + a_12 * b_21 + a_13 * b_31
- The element in the first row and second column of AB is given by:
AB_12 = a_11 * b_12 + a_12 * b_22 + a_13 * b_32
- The element in the first row and third column of AB is given by:
AB_13 = a_11 * b_13 + a_12 * b_23 + a_13 * b_33
- Similarly, calculate the remaining elements of AB.
3. Now, let's substitute the corresponding values from matrices A and B to calculate the elements of AB. We have:
AB_11 = (-5 * -2) + (4 * 8) + (-6 * 5)
AB_12 = (-5 * 4) + (4 * 9) + (-6 * -2)
AB_13 = (-5 * -5) + (4 * 3) + (-6 * 2)
AB_21 = (9 * -2) + (-8 * 8) + (7 * 5)
AB_22 = (9 * 4) + (-8 * 9) + (7 * -2)
AB_23 = (9 * -5) + (-8 * 3) + (7 * 2)
AB_31 = (4 * -2) + (-3 * 8) + (9 * 5)
AB_32 = (4 * 4) + (-3 * 9) + (9 * -2)
AB_33 = (4 * -5) + (-3 * 3) + (9 * 2)
4. Calculating these values, we find:
AB_11 = 2 + 32 - 30 = 4
AB_12 = -20 + 36 + 12 = 28
AB_13 = 25 + 12 - 12 = 25
AB_21 = -18 - 64 + 35 = -47
AB_22 = 36 - 72 - 14 = -50
AB_23 = -45 - 24 + 14 = -55
AB_31 = -8 - 24 + 45 = 13
AB_32 = 16 - 27 - 18 = -29
AB_33 = -20 - 9 + 18 = -11
5. Hence, matrix AB is:
AB = [ 4 28 25 ]
[-47 -50 -55 ]
[ 13 -29 -11 ]