Guided Example #1 - Matrices - Basics
Find the product. [31-12] [3250]
a. [11670]
b. [14674]
c. [1467-2]
d. Not possible
The correct answer is C. To see why, view each of the steps below.
Step 1
Important: We can only multiply matrices if the number of columns in the first matrix
is the same as the number of rows in the second matrix.
Step 2
Examples
Multiplying a 2 × 3 matrix by a 3 × 1 matrix is possible and it gives a 2 × 1 matrix as the answer.
Multiplying a 4 × 1 matrix by a 1 × 4 matrix is okay; it gives a 4 × 4 matrix.
A 3 × 3 matrix times a 2 × 3 matrix is NOT possible.
Step 3
We multiply and add the elements as follows. We work across the 1st row of the first matrix, multiplying down the 1st column of the second matrix, element by element. We add the resulting products. Our answer goes in position a11 (top left) of the answer matrix. [31-12][3250] = [3(3)+1(5)]
Step 4
We do a similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result is placed in position a12(top right). [31-12] [3250]= [3(3)+1(5)3(2)+1(0)]
Step 5
Now for the 2nd row of the first matrix and the 1st column of the second matrix. The result is placed in position a21.
[31-12] [3250]= [3(3)+1(5)3(2)+1(0)-1(3)+2(5)]
Step 6
Finally, we do the 2nd row of the first matrix and the 2nd column of the second matrix. The result is placed in position a22 [31-12] [3250] = [3(3)+1(5)3(2)+1(0)-1(3)+2(5)-1(2)+2(0)] = [1467-2]
All Steps
Important: We can only multiply matrices if the number of columns in the first matrix
is the same as the number of rows in the second matrix.
Examples
Multiplying a 2 × 3 matrix by a 3 × 1 matrix is possible and it gives a 2 × 1 matrix as the answer.
Multiplying a 4 × 1 matrix by a 1 × 4 matrix is okay; it gives a 4 × 4 matrix.
A 3 × 3 matrix times a 2 × 3 matrix is NOT possible.
We multiply and add the elements as follows. We work across the 1st row of the first matrix, multiplying down the 1st column of the second matrix, element by element. We add the resulting products. Our answer goes in position a11 (top left) of the answer matrix. [31-12][3250] = [3(3)+1(5)]
We do a similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result is placed in position a12(top right). [31-12] [3250]= [3(3)+1(5)3(2)+1(0)]
Now for the 2nd row of the first matrix and the 1st column of the second matrix. The result is placed in position a21.
[31-12] [3250]= [3(3)+1(5)3(2)+1(0)-1(3)+2(5)]
Finally, we do the 2nd row of the first matrix and the 2nd column of the second matrix. The result is placed in position a22 [31-12] [3250] = [3(3)+1(5)3(2)+1(0)-1(3)+2(5)-1(2)+2(0)] = [1467-2]