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Adding and Subtracting Matrices

Sum corresponding elements of same-dimension matrices

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Adding and Subtracting Matrices

Using the movie theater example from the previous lesson, how could we determine how much more the theater charges at night for each ticket type?

Recall that for a matinee movie, the movie theater charges the following prices:

Kids: $5 Adults: $8 Seniors: $6

For the same movie at night, the theater charges the following prices:

Kids: $7 Adults: $10 Seniors: $8

Adding and Subtracting Matrices

If two matrices have the same dimensions, then they can be added or subtracted by adding or subtracting corresponding elements as shown below.

Addition:

\begin{align*}\begin{bmatrix} a & b\\ c & d\end{bmatrix}+\begin{bmatrix} e & f\\ g & h\end{bmatrix}=\begin{bmatrix} a+e & b+f\\ c+g & d+h\end{bmatrix}\end{align*}

Subtraction:

\begin{align*}\begin{bmatrix} a & b\\ c & d\end{bmatrix}-\begin{bmatrix} e & f\\ g & h\end{bmatrix}=\begin{bmatrix} a-e & b-f\\ c-g & d-h\end{bmatrix}\end{align*}

It is important to note that the two matrices are not required to be square matrices. The requirement is that they are the same dimensions. In other words, you can add two matrices that are both \begin{align*}2 \times 3\end{align*}, but you cannot add a \begin{align*}2 \times 2\end{align*} matrix with a \begin{align*}3 \times 2\end{align*} matrix. Before attempting to add two matrices, check to make sure that they have the same dimensions.

Investigation: Commutative and Associative Properties of Addition

The Commutative Property of Addition states that \begin{align*}a + b = b + a\end{align*} for real numbers, \begin{align*}a\end{align*} and \begin{align*}b\end{align*}. Does this property hold for matrices? The Associative Property of Addition states that \begin{align*}a + (b + c) = (a + b) + c\end{align*} for real numbers, \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*}. Does this property hold for matrices? Consider the matrices below:

\begin{align*}A = \begin{bmatrix} -3 & 7\\ 4 & -1 \end{bmatrix} \qquad B = \begin{bmatrix} 5 & 1\\ -8 & -2 \end{bmatrix} \qquad C = \begin{bmatrix} -6 & -10\\ 5 & 3 \end{bmatrix}\end{align*}

1. Find \begin{align*}A + B \Rightarrow \begin{bmatrix} -3 & 7\\ 4 & -1 \end{bmatrix} + \begin{bmatrix} 5 & 1\\ -8 & -2 \end{bmatrix} = \begin{bmatrix} 2 & 8\\ -4 & -3 \end{bmatrix}\end{align*}

2. Find \begin{align*}B + A \Rightarrow \begin{bmatrix} 5 & 1\\ -8 & -2 \end{bmatrix} + \begin{bmatrix} -3 & 7\\ 4 & -1 \end{bmatrix} = \begin{bmatrix} 2 & 8\\ -4 & -3 \end{bmatrix}\end{align*}

Since \begin{align*}A + B = B + A\end{align*}, we can conjecture that matrix addition is commutative.

3. Find \begin{align*}(A + B) + C \Rightarrow \left( \begin{bmatrix} -3 & 7\\ 4 & -1 \end{bmatrix} + \begin{bmatrix} 5 & 1\\ -8 & -2 \end{bmatrix} \right) + \begin{bmatrix} -6 & -10\\ 5 & 3 \end{bmatrix} = \begin{bmatrix} 2 & 8\\ -4 & -3 \end{bmatrix} + \begin{bmatrix} -6 & -10\\ 5 & 3 \end{bmatrix} = \begin{bmatrix} -4 & -2\\ 1 & 0 \end{bmatrix}\end{align*}

4. Find \begin{align*}A + (B + C) \Rightarrow \begin{bmatrix} -3 & 7\\ 4 & -1 \end{bmatrix} + \left( \begin{bmatrix} 5 & 1\\ -8 & -2 \end{bmatrix} + \begin{bmatrix} -6 & -10\\ 5 & 3 \end{bmatrix} \right) = \begin{bmatrix} -3 & 7\\ 4 & -1 \end{bmatrix} + \begin{bmatrix} -1 & -9\\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -4 & -2\\ 1 & 0 \end{bmatrix}\end{align*}

Since \begin{align*}(A + B) + C = A + (B + C)\end{align*}, we can conjecture that the associative property is true for matrix addition as well.

Properties of Matrix Addition

Commutative Property: \begin{align*}A + B = B + A\end{align*}

Associative Property: \begin{align*}(A + B) + C = A + (B + C)\end{align*}

\begin{align*}^*\end{align*}Note that these properties do not work with subtraction with real numbers. For example: \begin{align*}7 -5 \ne 5 - 7\end{align*}. Because they do not hold for subtraction of real numbers, they also do not work with matrix subtraction.

Solve the following problems

Find the sum: \begin{align*}\begin{bmatrix} 4 & -5 & 6\\ -3 & 7 & 9 \end{bmatrix} + \begin{bmatrix} -1 & 4 & 8\\ 0 & -3 & 12 \end{bmatrix} =\end{align*}

By adding the elements in corresponding positions we get:

\begin{align*}\begin{bmatrix} 4 & -5 & 6\\ -3 & 7 & 9 \end{bmatrix} + \begin{bmatrix} -1 & 4 & 8\\ 0 & -3 & 12 \end{bmatrix} = \begin{bmatrix} 4+-1 & -5+4 & 6+8\\ -3+0 & 7+-3 & 9+12 \end{bmatrix} = \begin{bmatrix} 3 & -1 & 14\\ -3 & 4 & 21 \end{bmatrix}\end{align*}

Find the difference: \begin{align*}\begin{bmatrix} -7\\ 6\\ -9\\ 10 \end{bmatrix} - \begin{bmatrix} -3\\ -2\\ 8\\ 15 \end{bmatrix} =\end{align*}

By subtracting the elements in corresponding positions we get:

\begin{align*}\begin{bmatrix} -7\\ 6\\ -9\\ 10 \end{bmatrix} - \begin{bmatrix} -3\\ -2\\ 8\\ 15 \end{bmatrix} = \begin{bmatrix} -7-(-3)\\ 6-(-2)\\ -9-8\\ 10-15 \end{bmatrix} = \begin{bmatrix} -4\\ 8\\ -17\\ -5 \end{bmatrix}\end{align*}

Perform the indicated operation: \begin{align*}\begin{bmatrix} -4 & 2\\ 5 & -3\\ 13 & 8 \end{bmatrix} + \begin{bmatrix} 7 & -1 & 0\\ -5 & 2 & 6 \end{bmatrix}\end{align*}

In this case the first matrix is \begin{align*}3 \times 2\end{align*} and the second matrix is \begin{align*}2 \times 3\end{align*}. Because they have different dimensions they cannot be added.

Examples

Example 1

Earlier, you were asked to determine how much more the theater charges at night for each ticket. 

We could organize the data into two separate matrices and subtract.

\begin{align*}\begin{bmatrix}7 & 10 & 8\end{bmatrix} - \begin{bmatrix}5 & 8 & 6\end{bmatrix} = \begin{bmatrix}2 & 2 & 2\end{bmatrix}\end{align*}

We can now easily see that the movie theater charges $2 more for each ticket type at night.

Perform the indicated operation.

Example 2
\begin{align*}\begin{bmatrix}3 & -7\end{bmatrix} + \begin{bmatrix}-1 & 8\end{bmatrix}\end{align*}


\begin{align*}\begin{bmatrix} 3 & -7 \end{bmatrix} + \begin{bmatrix} -1 & 8 \end{bmatrix} = \begin{bmatrix} 3+(-1) & -7+8 \end{bmatrix} = \begin{bmatrix} 2 & 1 \end{bmatrix}\end{align*}

Example 3
\begin{align*}\begin{bmatrix} 1\\ -5 \end{bmatrix} - \begin{bmatrix} 3 & -3\\ 4 & 1 \end{bmatrix}\end{align*}

These matrices cannot be subtracted because they have different dimensions.

Example 4
\begin{align*}\begin{bmatrix} 6 & -7\\ -11 & 5 \end{bmatrix} - \begin{bmatrix} -2 & 4\\ -3 & 9 \end{bmatrix}\end{align*}


\begin{align*}\begin{bmatrix} 6 & -7\\ -11 & 5 \end{bmatrix} - \begin{bmatrix} -2 & 4\\ -3 & 9 \end{bmatrix} = \begin{bmatrix} 6-(-2) & -7-4\\ -11-(-3) & 5-9 \end{bmatrix} = \begin{bmatrix} 8 & -11\\ -8 & -4 \end{bmatrix}\end{align*}

Review

Perform the indicated operation (if possible).

  1. .
\begin{align*}\begin{bmatrix} 2 & -1\\ 5 & 0 \end{bmatrix} + \begin{bmatrix} -6 & 0\\ 3 & -4 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 3 & -2\\ -5 & 1\\ 10 & 9 \end{bmatrix} - \begin{bmatrix} -2 & 7\\ 10 & -8\\ 7 & 5 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 4 \\ -2\\ 12\\ 7 \end{bmatrix} + \begin{bmatrix} -1 \\ 9 \\ -2 \\ 0 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} -1 & -4 & -1 & 12\\ 2 & 6 & 14 & 5\end{bmatrix} - \begin{bmatrix} -3 & 1\\ 7 & -6 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 4 & -1\end{bmatrix} + \begin{bmatrix} 0 & 5\end{bmatrix} - \begin{bmatrix} -12 & 3 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 3 & 5 \end{bmatrix} + \begin{bmatrix}-2 & -1 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 2\\ 7 \end{bmatrix} + \begin{bmatrix} -3 & 5 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 11 & 7 & -3\\ 9 & 15 & 8 \end{bmatrix} + \begin{bmatrix} 20 & -4 & 7\\ 1 & 11 & -13 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 25\\ 19\\ -5 \end{bmatrix} - \begin{bmatrix} 11\\ 20\\ -3 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} 2 & -5 & 3\\ 9 & 15 & 8\\ -1 & -4 & 6 \end{bmatrix} + \begin{bmatrix} -3 & 8 & -3\\ 11 & -6 & -7\\ 0 & 8 & 5 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} -3 & 2\\ 4 & -1 \end{bmatrix} - \begin{bmatrix} 6 & -11 & 13\\ 17 & 8 & 10 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} -5 & 2\\ 9 & -3 \end{bmatrix} + \begin{bmatrix} -3 & -5\\ 8 & 12 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\left( \begin{bmatrix} 5 & -2\\ -3 & 1 \end{bmatrix} + \begin{bmatrix} -8 & 5\\ 6 & 13 \end{bmatrix} \right) - \begin{bmatrix} -10 & 8\\ 9 & 1 \end{bmatrix}\end{align*}
  1. .
\begin{align*}\begin{bmatrix} -5 & 2\\ 11 & 3 \end{bmatrix} - \left( \begin{bmatrix} 8 & -2\\ 3 & 5 \end{bmatrix} + \begin{bmatrix} -12 & 3\\ -6 & 15 \end{bmatrix} \right)\end{align*}
  1. .
\begin{align*}\left( \begin{bmatrix} 22 & -7\\ 5 & 3\\ 11 & -8 \end{bmatrix} - \begin{bmatrix} -8 & 9\\ 15 & 12\\ 10 & -1 \end{bmatrix} \right) + \begin{bmatrix} 5 & 11\\ 17 & -3\\ -9 & 4 \end{bmatrix}\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 4.2. 

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