About Introduction to Matrices:
A matrix is an ordered array of numbers. The order of a matrix is stated as the number of rows by the number of columns. For example, a matrix with 3 rows and 4 columns has an order of 3 × 4. If we have a matrix A, then a13 would be the element in A that is in the first row and third column. Two matrices are equal if and only if they have the same order and each corresponding element is equal.
Test Objectives
- Demonstrate the ability to find the order of a matrix
- Demonstrate the ability to find a specific entry in a matrix
- Demonstrate the ability to find values that make two matrices equal
#1:
Instructions: State the order.
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}5 & -1 & 3\\ 3 & 7 & 0\end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-9 & 19 & 17\\ 2 & 3 & 0 \\ 1 & -1 & 5\end{array}\right]$$
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#2:
Instructions: State the order.
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccccc}5 & 2 & 19 & 13 & 22 \\ -17 & -4 & 8 & 29 & 1 \\ -9 & -7 & -3 & 44 & 50\end{array}\right]$$
Instructions: Find the indicated element.
$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}5 & 6 \\ 1 & -1 \\ 7 & 3\end{array}\right]$$ $$a_{12} = \,?$$
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#3:
Instructions: Find the indicated element.
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}5 & 6 \\ 1 & -1 \\ 7 & 3\end{array}\right]$$ $$a_{23} = \, ?$$
Instructions: Answer the given question.
b) What type of matrix has the same number of rows as columns?
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#4:
Instructions: If A = B, what is the value of x?
$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}1 & 5 \\ 6 & x\end{array}\right]$$ $$B=\left[ \begin{array}{cc}1 & 5 \\ 6 & 7\end{array}\right]$$
Instructions: If A = B, what is the value of x and y?
$$b)\hspace{.2em}$$
$$A=\left[ \begin{array}{cc}-3 & 1 \\ 6 & x + 5\end{array}\right]$$ $$B=\left[ \begin{array}{cc}y + 2 & 1 \\ 6 & -2\end{array}\right]$$Watch the Step by Step Video Solution View the Written Solution
#5:
Instructions: Find x, y, and z such that A = B.
$$a)\hspace{.2em}$$
$$A=\left[ \begin{array}{ccc}-1 & 5 & 9 \\ x - 1 & y + 2 & 4\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-1 & 5 & 9 \\ 13 & -21 & z + 4\end{array}\right]$$$$b)\hspace{.2em}$$
$$A=\left[ \begin{array}{ccc}3 & 5 & 11 \\ x + 1 & y - 7 & 7\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}3 & 11 & 5 \\ -14 & 12 & z - 9\end{array}\right]$$Watch the Step by Step Video Solution View the Written Solution
Written Solutions:
#1:
Solutions:
a) 2 × 3
b) 3 × 3
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#2:
Solutions:
a) 3 × 5
b) a12 = 6
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#3:
Solutions:
a) Doesn't exist
b) Square Matrix
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#4:
Solutions:
$$a)\hspace{.2em}x = 7$$
$$b)\hspace{.2em}x=-7, y=-5$$
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#5:
Solutions:
$$a)\hspace{.2em}x=14, y=-23, z=0$$
$$b)\hspace{.2em}A ≠ B$$ Matrices A and B can never be equal since: $$a_{12} = 5 \, \text{and} \, b_{12} = 11$$ $$a_{13} = 11 \, \text{and} \, b_{13} = 5$$ No values of x, y, or z can ever make A = B.
