Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in the table below.

Converting the data to a matrix, we have

[latex]{C}_{2013}=\left[\begin{array}{c}15\\ 16\\ 16\end{array}\begin{array}{c}27\\ 34\\ 34\end{array}\right][/latex]

To calculate how much computer equipment will be needed, we multiply all entries in matrix [latex]C[/latex] by 0.15.

[latex]\left(0.15\right){C}_{2013}=\left[\begin{array}{c}\left(0.15\right)15\\ \left(0.15\right)16\\ \left(0.15\right)16\end{array}\begin{array}{c}\left(0.15\right)27\\ \left(0.15\right)34\\ \left(0.15\right)34\end{array}\right]=\left[\begin{array}{c}2.25\\ 2.4\\ 2.4\end{array}\begin{array}{c}4.05\\ 5.1\\ 5.1\end{array}\right][/latex]

We must round up to the next integer, so the amount of new equipment needed is

[latex]\left[\begin{array}{c}3\\ 3\\ 3\end{array}\begin{array}{c}5\\ 6\\ 6\end{array}\right][/latex]

Adding the two matrices as shown below, we see the new inventory amounts.

[latex]\left[\begin{array}{c}15\\ 16\\ 16\end{array}\begin{array}{c}27\\ 34\\ 34\end{array}\right]+\left[\begin{array}{c}3\\ 3\\ 3\end{array}\begin{array}{c}5\\ 6\\ 6\end{array}\right]=\left[\begin{array}{c}18\\ 19\\ 19\end{array}\begin{array}{c}32\\ 40\\ 40\end{array}\right][/latex]

This means

[latex]{C}_{2014}=\left[\begin{array}{c}18\\ 19\\ 19\end{array}\begin{array}{c}32\\ 40\\ 40\end{array}\right][/latex]

Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If [latex]A[/latex] is an [latex]\text{ }m\text{ }\times \text{ }r\text{ }[/latex] matrix and [latex]B[/latex] is an [latex]\text{ }r\text{ }\times \text{ }n\text{ }[/latex] matrix, then the product matrix [latex]AB[/latex] is an [latex]\text{ }m\text{ }\times \text{ }n\text{ }[/latex] matrix. For example, the product [latex]AB[/latex] is possible because the number of columns in [latex]A[/latex] is the same as the number of rows in [latex]B[/latex]. If the inner dimensions do not match, the product is not defined.

We multiply entries of [latex]A[/latex] with entries of [latex]B[/latex] according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row [latex]i[/latex] of [latex]AB,\text{}[/latex] we multiply the entries in row [latex]i[/latex] of [latex]A[/latex] by column [latex]j[/latex] in [latex]B[/latex] and add. For example, given matrices [latex]A[/latex] and [latex]B,\text{}[/latex] where the dimensions of [latex]A[/latex] are [latex]2\text{ }\times \text{ }3[/latex] and the dimensions of [latex]B[/latex] are [latex]3\text{ }\times \text{ }3,\text{}[/latex] the product of [latex]AB[/latex] will be a [latex]2\text{ }\times \text{ }3[/latex] matrix.

[latex]A=\left[\begin{array}{rrr}\hfill {a}_{11}& \hfill {a}_{12}& \hfill {a}_{13}\\ \hfill {a}_{21}& \hfill {a}_{22}& \hfill {a}_{23}\end{array}\right]\text{ and }B=\left[\begin{array}{rrr}\hfill {b}_{11}& \hfill {b}_{12}& \hfill {b}_{13}\\ \hfill {b}_{21}& \hfill {b}_{22}& \hfill {b}_{23}\\ \hfill {b}_{31}& \hfill {b}_{32}& \hfill {b}_{33}\end{array}\right][/latex]

Multiply and add as follows to obtain the first entry of the product matrix [latex]AB[/latex].

1. To obtain the entry in row 1, column 1 of [latex]AB,\text{}[/latex] multiply the first row in [latex]A[/latex] by the first column in [latex]B[/latex], and add.
   [latex]\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{11}\\ {b}_{21}\\ {b}_{31}\end{array}\right]={a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}[/latex]
2. To obtain the entry in row 1, column 2 of [latex]AB,\text{}[/latex] multiply the first row of [latex]A[/latex] by the second column in [latex]B[/latex], and add.
   [latex]\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{12}\\ {b}_{22}\\ {b}_{32}\end{array}\right]={a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}[/latex]
3. To obtain the entry in row 1, column 3 of [latex]AB,\text{}[/latex] multiply the first row of [latex]A[/latex] by the third column in [latex]B[/latex], and add.
   [latex]\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\end{array}\right]\cdot \left[\begin{array}{c}{b}_{13}\\ {b}_{23}\\ {b}_{33}\end{array}\right]={a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}[/latex]

We proceed the same way to obtain the second row of [latex]AB[/latex]. In other words, row 2 of [latex]A[/latex] times column 1 of [latex]B[/latex]; row 2 of [latex]A[/latex] times column 2 of [latex]B[/latex]; row 2 of [latex]A[/latex] times column 3 of [latex]B[/latex]. When complete, the product matrix will be

[latex]AB=\left[\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{11}+{a}_{12}\cdot {b}_{21}+{a}_{13}\cdot {b}_{31}\\ \end{array}\\ {a}_{21}\cdot {b}_{11}+{a}_{22}\cdot {b}_{21}+{a}_{23}\cdot {b}_{31}\end{array}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{12}+{a}_{12}\cdot {b}_{22}+{a}_{13}\cdot {b}_{32}\\ \end{array}\\ {a}_{21}\cdot {b}_{12}+{a}_{22}\cdot {b}_{22}+{a}_{23}\cdot {b}_{32}\end{array}\begin{array}{c}\begin{array}{l}{a}_{11}\cdot {b}_{13}+{a}_{12}\cdot {b}_{23}+{a}_{13}\cdot {b}_{33}\\ \end{array}\\ {a}_{21}\cdot {b}_{13}+{a}_{22}\cdot {b}_{23}+{a}_{23}\cdot {b}_{33}\end{array}\right][/latex]

Example: Multiplying Two Matrices

Multiply matrix [latex]A[/latex] and matrix [latex]B[/latex].

[latex]A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\text{ and }B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right][/latex]

Solution

First, we check the dimensions of the matrices. Matrix [latex]A[/latex] has dimensions [latex]2\times 2[/latex] and matrix [latex]B[/latex] has dimensions [latex]2\times 2[/latex]. The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions [latex]2\times 2[/latex].

We perform the operations outlined previously.