**Scalar Multiplication and Matrix to Matrix Multiplication**

There are two types or categories where matrix multiplication usually falls under. The first one is called **Scalar Multiplication**, also known as the “**Easy Type**“; where you simply multiply a single number into each and every entry of a given matrix.

The second type is called **Matrix to Matrix Multiplication** known as the “**Messy Type**“.

Let’s compare the two.

**Scalar Multiplication**

This is the “Easy Type” because the approach is extremely simple or straightforward.

**Description: **Take the number outside the matrix (known as scalar) and multiply it to each and every entry or element of the matrix.

**Matrix-to-Matrix Multiplication**

This is the “Messy Type” because the process is very much involved. However, you will realize later after going through the procedure and some examples that the steps required are manageable. Don’t worry, I will help you with this!

The two matrices being multiplied **must** have the following characteristic:

The number of columns of the **left matrix** must equal the number of rows of the **right matrix**.

First, let’s look at Scalar Multiplication…

**Directions**: Given the following matrices, perform the indicated operation. Apply **scalar multiplication** as part of the overall simplification process.

**Example 1**: Perform the indicated operation for **2 A**.

I will take the scalar 2 (similar to the coefficient of a term) and distribute by multiplying it to each entry of matrix *A*. In case you forgot, you may review the generic formula above.

Since matrix *A* is

then **2 A** is solved by…

That’s all there is to it. Done!

**Example 2**: Perform the indicated operation for **–****3 B**.

I will do the same thing similar to Example 1. No big deal! Multiply the negative scalar, −3, into each element of matrix *B*.

Since matrix *B* is

then matrix** −3 B** is solved by…

Did you arrive at the same final answer? If not, please recheck your work to make sure that it matches with the correct answer.

**Example 3**: Perform the indicated operation for **–2 D + 5F**.

To solve this problem, I need to apply scalar multiplication twice and then add their results to get the final answer.

- First, find the value of matrix
**−2***D*

I know that matrix *D* is

Therefore, −2*D* is obtained as follows using scalar multiplication.

- Second, find the value of
**5***F*

Matrix *F* is given as

That means 5*F* is solved using scalar multiplication.

- Now, I can solve for –2
*D*+ 5*F*by adding the values of matrices**−2**and*D***5**, as shown above. Click here to review how to add and subtract matrices.*F*

That’s it!

## Matrix to Matrix Multiplication a.k.a "Messy Type"

The next type of problem is called **Matrix Multiplication**. This is the “Messy Type” because you need to follow a certain procedure in order to get it right. But first, we need to ensure that the two matrices are “allowed” to be multiplied together. Otherwise, the given two matrices are “incompatible” to be multiplied. If this is the case, we say that the solution is undefined.

**Always remember this !**

An example of two matrices that are “incompatible” or cannot be multiplied together is the following…

Suppose we’re given matrices A and B, find *AB* (do matrix multiplication, if applicable). Determine which one is the left and right matrices based on their location. It is a very important step.

To determine if I can multiply the two given matrices, I need to pay attention on the number of columns of matrix A and the number of rows of matrix B. If they are equal, then I can proceed with Matrix Multiplication. Otherwise, I will conclude that the answer is undefined!

Because **Matrix A has the number of columns of 2**, and **Matrix B has the number of rows of 3**, and they are not equal (2 ≠ 3) , I conclude that ** AB = undefined**. That means their product can’t be found.

**Directions**: Given the following matrices, perform the indicated operation which is to find their product.

**Example 1**: Calculate, if possible, the product of ** BE**.

In order for matrices B and E to have a product, the number of columns of left matrix B **must equal** to the number of rows of right matrix E.

**Matrix B (left)**

number of columns = 3

**Matrix E (right)**

number of rows = 3

Since this is the case, then it is okay to multiply them together. Now, these are the steps:

**Step 1:** Place them side by side.

**Step 2:** Multiply the rows of B into the columns of E by multiplying the corresponding elements of each row to each elements of column, and then add them together.

**Please watch the animated solution carefully.**

If you have no patience watching the animated solution above on how to perform matrix multiplication, you can view the regular solution I have included below.

**Example 2**: Calculate, if possible, the product of * EF*.

Check first if the product of the two matrices exists by making sure that the number of columns of left matrix E **equals** the number of rows of right matrix F.

**Matrix E (left)**

number of columns = 2

**Matrix F (right)**

number of rows = 2

This is wonderful since the **number of columns of matrix E equals the number of rows of matrix F**. This means the product of

*is defined so we can go ahead and perform matrix multiplication. See below for the animated step by step solution of matrix multiplication.*

**EF****Example 3**: Calculate, if possible, the product of * FE*.

In our previous example, we have successfully obtained the product of *EF*. This time around, we want to find if we can find the product of *E* and *F*, in that order.

Just to remind you, real numbers are commutative under multiplication operation which means that the order of multiplication does not affect the final product. For instance…

So the big question becomes, does it work also in matrix multiplication?

Let’s check if the number of columns of matrix F equals the number of rows of matrix E.

**Matrix F (left)**

number of columns = 2

**Matrix E (right)**

number of rows = 3

Obviously, the number of columns of Matrix F **does not equal** the number of rows of Matrix E. The implication is that the product of *FE* cannot be calculated, therefore undefined!

In general, matrix multiplication is not commutative.

**Example 4**: Calculate, if possible, the product of * AE*.

The standard way to describe the size or dimension of a matrix is to…

(state number of rows) **x** (state number of columns)

…read as “the number of rows by the number of columns”.

3 **x **3 (three by three matrix)

3 **x **2 (three by two matrix)

Since the **number of columns of matrix A** equals the **number of rows of matrix E** then we conclude that the product of *AE* is defined.

Let’s work it out. See animated solution below.

**Example 5**: Calculate, if possible, the product of ** EA**.

3 **x **2 (three by two matrix)

3 **x **3 (three by three matrix)

Obviously, the number of columns of matrix E **does not equal** the number of columns of matrix A. Therefore, the product of *EA* cannot be calculated, or undefined.

**Example 6**: Calculate, if possible, the product of ** DF**.

Try this out on paper first then click below to compare your answer.

**Solution to Example 6:**

Since the number of columns of matrix D **equals** the number of rows of matrix F, the product of *DF* is defined.