Adding and Subtracting Matrices

Matrices arise in many branches of mathematics. If you''re studying pre-calculus material, it''s a safe bet that you will encounter matrices at some point. A matrix is simply a table of numerical information with columns and rows of numbers. Matrices are useful in a variety of real-world contexts. Matrices store data, which makes them particularly useful in computer science and graphics. We also use matrices if we need to examine quantum systems and the behavior of subatomic particles. Before learning how to add and subtract matrices, it''s advisable to brush up on related matrix topics like matrix dimensions, the associative properties of matrices, and solving matrix equations.

How to add and subtract matrices

We can add and subtract matrices in much the same way that we do with ordinary numbers. Imagine we have two separate matrices, matrix A and matrix B.

$A=\left[\begin{array}{ccc}3& 5& 2\\ 2& 7& 4\\ 4& -1& 4\end{array}\right]$

We might be given a problem that asks us to add matrices A and B. In such a case, we add each corresponding entry like so:

$A+B=\left[\begin{array}{ccc}3+2& 5+\left(-5\right)& 2+3\\ 2+3& 7+4& 4+1\\ 4+3& -1+5& 4+5\end{array}\right]=\left[\begin{array}{ccc}5& 0& 5\\ 5& 11& 5\\ 7& 4& 9\end{array}\right]$

It''s the same simple addition we learned at the very beginning of school. The only difference here is that we''re working with matrix entries.

Let''s try one with subtraction. It''s just as easy as addition.

Matrix C is

$\left[\begin{array}{ccc}6& 2& 8\\ 3& 3& 7\end{array}\right]$

Matrix D is

$\left[\begin{array}{ccc}2& 3& 5\\ 1& 0& 3\end{array}\right]$

Find $C-D$ .

$C-D=\left[\begin{array}{ccc}6-2& 2-3& 8-5\\ 3-1& 3-0& 7-3\end{array}\right]=\left[\begin{array}{ccc}4& -1& 3\\ 2& 3& 4\end{array}\right]$

Notice that we cannot add or subtract matrices of different dimensions. For instance, we couldn''t add a $3\times 3$ matrix to a $2\times 2$ matrix because there''s a dimensional mismatch. Each matrix entry needs a corresponding value to add or subtract, so trying to perform these kinds of operations simply makes no sense.

Matrix addition and subtraction practice problems

Now that we''ve seen how to add and subtract matrices, let''s try some practice problems to solidify our understanding of the topic.

1. Given that $A=\left[\begin{array}{ccc}\frac{1}{3}& 4& -7\\ -2& 8& 10\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 6& \frac{2}{3}\\ 0& -7& 3\end{array}\right]$ , find $A+B$ .
   

   $A+B=\left[\begin{array}{ccc}\frac{4}{3}& 10& \frac{-19}{3}\\ -2& 1& 13\end{array}\right]$

2. Given that $C=\left[\begin{array}{ccc}7& 9& -5\\ 2& -9& -7\\ 5& -10& 1\end{array}\right]$ and $D=\left[\begin{array}{ccc}-7& 3& 2\\ -8& 10& -6\\ 5& \frac{1}{4}& 10\end{array}\right]$ , find $C-D$ .

   $C-D=\left[\begin{array}{ccc}14& 6& -3\\ 10& -19& -1\\ 0& -\frac{41}{4}& -9\end{array}\right]$

3. Given that $A=\left[\begin{array}{ccc}-6& 7& 2\\ -8& -3& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-6& 0& 5\\ -1& -8& -1\end{array}\right]$ , find $A+B$ .
   

   $A+B=\left[\begin{array}{ccc}-12& 7& 7\\ -9& 5& 0\end{array}\right]$

4. Given that $E=\left[\begin{array}{cc}-9& 10\\ 7& 10\\ -1& -4\end{array}\right]$ and $F=\left[\begin{array}{cc}-5& -8\\ 10& -9\\ 2& -2\end{array}\right]$ , find $E-F$ .

   $E-F=\left[\begin{array}{cc}-4& 18\\ -3& 19\\ -3& 2\end{array}\right]$

5. Given that $A=\left[\begin{array}{ccc}-\frac{3}{2}& 2& 0\\ -9& -\frac{1}{3}& -4\\ 1& -\frac{5}{6}& 6\end{array}\right]$ and $B=\left[\begin{array}{ccc}3& \frac{5}{9}& 8\\ -\frac{1}{2}& -4& -6\\ \frac{5}{6}& 3& -\frac{1}{9}\end{array}\right]$ , find $A+B$ .

   $A+B=\left[\begin{array}{ccc}-\frac{9}{2}& \frac{23}{9}& 8\\ -\frac{19}{2}& -\frac{13}{3}& -10\\ \frac{11}{6}& \frac{13}{6}& \frac{53}{9}\end{array}\right]$

6. Given that $A=\left[\begin{array}{ccc}-5& -2& 8\\ -10& 9& -1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-10& -2& 8\\ 0& -3& 1\\ 8& 0& 10\end{array}\right]$ , find $A-B$ .
   This operation is not possible because the matrices have different dimensions.

Topics related to the Adding and Subtracting Matrices

Matrix Dimensions

Solving Matrix Equations

Associative Property of Matrices

Flashcards covering the Adding and Subtracting Matrices

Precalculus Flashcards

CLEP Precalculus Flashcards

Practice tests covering the Adding and Subtracting Matrices

Precalculus Diagnostic Tests

Get assistance learning how to add and subtract matrices

If you''re struggling with matrices, help is available. A tutor can demonstrate the process of adding and subtracting matrices, walk you through practice problems, and respond to any concerns that you have. If you have questions, they will assist you in your search for answers. In addition, if you have a student in high school who is studying matrices, a tutor will help them as well. It''s important to have a good grasp of matrix addition and subtraction, especially if you plan on pursuing more advanced mathematical coursework. Knowing these basic matrix operations will form a solid foundation for working with matrices in more advanced ways. Reach out to an Educational Director at Varsity Tutors in order to get started with a high-quality tutor.