Compatible Matrices

When adding and subtracting matrices, each matrix must have the same dimensions. But when multiplying matrices, they must instead be compatible, which means the number of columns in the first matrix must be equal to the number of rows in the second matrix. Let''s take a closer look at compatible matrices.

Compatible matrices: Definition and multiplication

Compatible matrices are matrices that can be multiplied; the number of columns in the first matrix equals the number of rows in the second matrix.

Let''s look at the following examples:

Example #1

Matrix $A$ is:

$\left[\begin{array}{ccc}2& 1& 3\end{array}\right]$

Matrix $B$ is:

$\left[\begin{array}{cc}2& 1\\ 3& 4\\ 1& 2\end{array}\right]$

In this example, the number of columns in the first matrix (3) equals the number of rows in the second matrix (3), so we know these two matrices are compatible and can be multiplied.

Example #2

Matrix $A$ is:

$\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$

Matrix $B$ is:

$\left[\begin{array}{cc}2& 1\\ 3& 4\\ 1& 2\end{array}\right]$

In this example, the number of columns in the first matrix (2) does not equal the number of rows in the second matrix (3). Therefore, these matrices are not compatible.

What if we''d like to multiply the compatible matrices in Example #1?

First, we need to find the dimensions of the product. The product of an $\left(a\times b\right)$ matrix and $a\left(b\times c\right)$ matrix has dimensions $\left(a\times c\right)$ . So, in this case, the product of the $1\times 3$ matrix (Matrix $A$ ) and the $3\times 2$ matrix (Matrix $B$ ) has dimensions $1\times 2$ .

Now, let''s multiply the matrices.

$A\times B=\left[\begin{array}{ccc}2& 1& 3\end{array}\right]\times \left[\begin{array}{cc}2& 1\\ 3& 4\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}2\times 2+1\times 3+3\times 1& 2\times 1+1\times 4+3\times 2\end{array}\right]=\left[\begin{array}{cc}10& 12\end{array}\right]$

So, the product of Matrix $A$ and Matrix $B$ is a $1\times 2$ matrix: $\left[\begin{array}{cc}10& 12\end{array}\right]$ .

Practice questions on compatible matrices

1. True or false: For compatible matrices, the number of columns in the first matrix equals the number of rows in the second matrix.
   Answer: True
2. What are the dimensions of the product of a $2\times 4$ matrix and a $4\times 7$ matrix?
   Answer: $2\times 7$
3. Is a $4\times 6$ matrix compatible with a $5\times 4$ matrix?
   Answer: No, the number of columns in the first matrix does not equal the number of rows in the second matrix.
4. Is a $3\times 5$ matrix compatible with a $5\times 7$ matrix?
   Answer: Yes, the number of columns in the first matrix equals the number of rows in the second matrix.
5. Multiply the following compatible matrices:

   $\left[\begin{array}{ccc}1& 3& 2\end{array}\right]\times \left[\begin{array}{c}5\\ 2\\ 1\end{array}\right]$

   The matrices have dimensions of $1\times 3$ and $3\times 1$ , so the dimensions of the product are $1\times 1$ .
   Multiply the corresponding numbers for the row and column. Add the products. $1\left(5\right)+3\left(2\right)+2\left(1\right)=5+6+2=13$ Answer: $\left[\begin{array}{c}13\end{array}\right]$

Topics related to the Compatible Matrices

Matrix Multiplication

Multiplying Vector by a Matrix

Matrix Dimensions

Flashcards covering the Compatible Matrices

Precalculus Flashcards

CLEP Precalculus Flashcards

Practice tests covering the Compatible Matrices

Precalculus Diagnostic Tests

Gain greater insight into compatible matrices

Compatible matrices can be challenging to grasp, and understanding how to multiply matrices can be even more difficult when doing so for the first time. Working alongside a tutor is a great way to gain clarity. A qualified personal educator can answer any questions whether about finding the dimensions of the product or determining which numbers in each row and column should be multiplied. Tutors can also provide valuable assistance with assignments and test preparations. Get more information about the benefits of tutoring by contacting the Educational Directors at Varsity Tutors today.