Multiplying Two Matrices

**Matrix Multiplication**

**Multiplying Two Matrices**

To find the product of two matrices, such as matrix

*M*and*N*, we need to look at the first row for matrix*M*and the first column for matrix*N*.Once you find them, multiply corresponding elements, and find the sum of their products. You could interpret this as finding the dot product of the row and column.

(1 × 7) + (2 × 9) + (3 × 11) =

**59**This is element

*a*_{11}of the resultant matrix.Now locate the first row for matrix

*M*and the second column for matrix*N*. How can you find the second element for the resultant matrix?Once you find them, multiply corresponding elements, and find the sum of their products.

(1 × 8) + (2 × 10) + (3 × 12) =

**64**This is element

*a*_{12}of the resultant matrix.For the next step we are going to repeat the same procedure, but this time we will work with the second row of

*M*and the first column of*N*, followed by locating the second row of*M*and second column of*N,*and we obtain(4 × 7) + (5 × 9) + (6 × 11) =

**132**This is element

*a*_{21}of the resultant matrix.and

(4 × 8) + (5 × 10) + (6 × 12) =

**154**This is element

*a*_{22}of the resultant matrix.Therefore, the resultant matrix is

**Can These Matrices Be Multiplied Together?**

If

*M*is an*x*×*y*matrix, and*N*is a*y*×*z*matrix, then the product*M**N*is an*x*×*z*matrix. Not all matrices can be multiplied together; to determine whether matrix*M*and*N*can be multiplied we need to look at the size of each matrix. The number of columns in*M*and the number of rows in*N*must be the same. Then you can multiply the matrices, and the values of*x*and*z*will determine the size of the resultant.*M**

*N*=

*MN*

*x*×

*y * y*×

*z*=

*x*×

*z*

So, if matrix

*A*is 3 × 4 and matrix*B*is a 4 × 2, then*AB**is possible since 4 = 4, and the size of the resultant matrix is 3 × 2**.*

*BA*

*is not possible in this case because the number of columns in*

*B*is not the same as the number of rows in

*A*(2 ≠ 3).

not possible

It is important to know that matrix multiplication is not commutative. Even if it's possible to multiply matrices

*M*and*N*in both directions,*MN*≠*NM*.**Practice**

Multiply if possible.

1)

2)

3)

4)

**Answers**

1)

2)

3) Not possible

4)