About Adding and Subtracting Matrices:
To add two or more matrices together, they need to have the same order. If the matrices being added all have the same order, we can find the sum by adding all corresponding elements position by position. Similarly, if we are dealing with subtracting one matrix from another, the two matrices must have the same order. If the two matrices have the same order, we can find the difference by subtracting all corresponding elements position by position.
Test Objectives
- Demonstrate the ability to add matrices
- Demonstrate the ability to subtract matrices
- Demonstrate the ability to solve a simple matrix equation
#1:
Instructions: Find A + B.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}3 & 1\\ 5 & 7\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-1 & -4\\ 5 & 6\end{array}\right]$$
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}-3 & 14\\ 18 & -1\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-7 & 8\\ 13 & 11\end{array}\right]$$
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#2:
Instructions: Find A + B.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-2 & 4 & 5\\ 8 & 1 & 6 \\ 12 & 19 & 5\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-2 & 1 & 5\\ 9 & 4 & -3 \\ 5 & 8 & 6\\ -1 & 7 & 6\end{array}\right]$$
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}1 & 9 & -2\\ -6 & 3 & -5 \\ 27 & -20 & 11\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-1 & -9 & 2\\ 6 & -3 & 5 \\ -27 & 20 & -11\end{array}\right]$$
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#3:
Instructions: Find A - B.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}8 & 3\\ 2 & -4\\ 1 & -11\end{array}\right]$$ $$B=\left[ \begin{array}{cc}1 & 10\\ 4 & 9\\ -2 & 11\end{array}\right]$$
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-1 & -3 & -4\\ 20 & 5 & 7 \\ 19 & 15 & 27\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}9 & 10 & 12\\ 5 & 1 & 8 \\ 3 & 6 & 5\end{array}\right]$$
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#4:
Instructions: Find A - B.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}3 & 0 & 1\\ -1 & 8 & 0 \\ -99 & 13 & 7\end{array}\right]$$ $$B=\left[ \begin{array}{cccc}4 & 12 & 8 & -1\\ -5 & 10 & 9 & -2 \\ -2 & 14 & 1 & -3\end{array}\right]$$
Instructions: A + X = B, find X.
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}-2 & 4\\ 17 & 5\end{array}\right]$$ $$B=\left[ \begin{array}{cc}3 & -5\\ 8 & 7\end{array}\right]$$
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#5:
Instructions: A + X = B, find X.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}-5 & -14 & 13\\ 9 & -7 & 11\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-1 & 2\\ 3 & 4\\ 6 & -9\end{array}\right]$$
Instructions: A - X = B, find X.
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-3 & 7 & 2\\ -2 & 0 & -1 \\ 3 & 4 & 0\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}15 & 19 & 0\\ 14 & -7 & -1\\ 22 & 13 & 8\end{array}\right]$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}2 & -3\\ 10 & 13\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}-10 & 22\\ 31 & 10\end{array}\right]$$
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#2:
Solutions:
a) The sum A + B doesn't exist.
Since A is a 3 × 3 and B is a 4 × 3, the addition operation is not possible.
b) 03 × 3
Note: "O" is sometimes used to represent the zero matrix, which is a matrix whose elements are all equal to zero. The notation can vary and you might see the number zero "0" used as well. The subscript notation (3 × 3 in this case) is sometimes used to indicate the order of the zero matrix. $$O_{3 × 3} = \left[ \begin{array}{ccc}0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$$ Written with "0" instead: $$0_{3 × 3} = \left[ \begin{array}{ccc}0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$$
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#3:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}7 & -7\\ -2 & -13 \\ 3 & -22\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-10 & -13 & -16\\ 15 & 4 & -1 \\ 16 & 9 & 22\end{array}\right]$$
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#4:
Solutions:
a) The difference A - B doesn't exist.
Since A is a 3 × 3 and B is a 3 × 4, the subtraction operation is not possible.
$$b)\hspace{.2em}$$ $$X=\left[ \begin{array}{cc}5 & -9\\ -9 & 2\end{array}\right]$$
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#5:
Solutions:
a) No Solution
$$A + X = B$$ Solving the matrix equation for X gives us: $$X = B - A$$ Since A is a 2 × 3 and B is a 3 × 2, the subtraction operation is not possible. This means there is no solution to our equation since there is no matrix X that is the result of B - A.
$$b)\hspace{.2em}$$ $$X=\left[ \begin{array}{ccc}-18 & -12 & 2\\ -16 & 7 & 0 \\ -19 & -9 & -8\end{array}\right]$$
