Matrices on the ACT -5 Matrix Problems
Instructions: You will see at least one question on matrices on the ACT. Complete the ACT matrix problems below. The answers are provided in the next section. You may want to review the examples and explanations in the last section of this page before you try this exercise.
1) What is Matrix A + Matrix B?
$ \text {Matrix A} \begin{bmatrix} 3 & 4 \\ 1 & 5 \end{bmatrix} \text {Matrix B} \begin{bmatrix} 6 & 2 \\ 7 & 8 \end{bmatrix} $
2) What is Matrix P – Matrix Q?
$ \text {Matrix P} \begin{bmatrix} 10 & 8 \\ -1 & 17 \end{bmatrix} \text {Matrix Q} \begin{bmatrix} -5 & 12 \\ -4 & 9 \end{bmatrix} $
3) Solve the following:
$ 3 \times \begin{bmatrix} 5 & 9 \\ 11 & 7 \end{bmatrix} $ = ?
4) What is the determinant of the following matrix?
$ \begin{bmatrix} 2 & 16 \\ 4 & 24 \end{bmatrix} $
5) What is the determinant of the following matrix?
$ \begin{bmatrix} -5 & 4\\ -6 & 9 \end{bmatrix} $
Matrices on the ACT – Answers to the Matrix Problems
Answer 1
1) Add the numbers from Matrix A to those in the same position in Matrix B, as shown below.
$ \text {Matrix A} \begin{bmatrix} 3 & 4 \\ 1 & 5 \end{bmatrix} \text {+ Matrix B} \begin{bmatrix} 6 & 2 \\ 7 & 8 \end{bmatrix}$ =
$ \begin{bmatrix} 3 + 6 & 4 + 2 \\ 1 + 7 & 5 + 8\end{bmatrix}$ = $ \begin{bmatrix} 9 & 6 \\ 8 & 13\end{bmatrix} $
Answer 2
Subtract the numbers from Matrix Q from those in the same position in Matrix P, as shown below.
$ \text {Matrix P} \begin{bmatrix} 10 & 8 \\ -1 & 17 \end{bmatrix} \text { – Matrix Q} \begin{bmatrix} -5 & 12 \\ -4 & 9 \end{bmatrix} $ =
$ \begin{bmatrix} 10 – -5 & 8 – 12 \\ -1 – -4 & 17 – 9\end{bmatrix}$ = $ \begin{bmatrix} 15 & -4 \\ 3 & 8\end{bmatrix} $
Answer 3
Multiply each number by 3 to solve:
$ 3 \times \begin{bmatrix} 5 & 9 \\ 11 & 7 \end{bmatrix}$ = $ \begin{bmatrix} 5 \times 3 & 9 \times 3 \\ 11 \times 3 & 7 \times 3\end{bmatrix} $ = $ \begin{bmatrix} 15 & 27 \\ 33 & 21 \end{bmatrix}$
Answer 4
To find the determinant, you need to cross multiply to get two products. Then subtract these two products to get the determinant.
$ \begin{bmatrix} 2 & 16 \\ 4 & 24 \end{bmatrix} $
(2 × 24) – (4 × 16) = 48 – 64 = -16
Answer 5
Be careful with the negative numbers when multiplying and adding.
$ \begin{bmatrix} -5 & 4\\ -6 & 9 \end{bmatrix} $
(-5 × 9) – (-6 × 4) = -45 – -24 = -21
What are Matrices?
The matrices that you will see on the ACT math exam will normally be in a two-by-two format like the one below.
$ \begin{bmatrix} a & b \\ c & d \end{bmatrix}$This means that these types of matrices are represented in a box-like format, consisting of 4 numbers.
Two numbers will be at the top of the matrix, and two numbers will be directly below these on the bottom of the matrix.
Performing Operations on Matrices – Examples
Example 1 – Adding Matrices on the ACT
1) Two add two matrices together, you need to add the numbers from the first matrix to those in the same position in second matrix
$ \text {Matrix A} \begin{bmatrix} 8 & 3 \\ 7 & 6 \end{bmatrix} \text {+ Matrix B} \begin{bmatrix} 2 & 5 \\ 9 & 1 \end{bmatrix}$ =
$ \begin{bmatrix} 8 + 2 & 3 + 5 \\ 7 + 9 & 6 + 1\end{bmatrix}$ = $ \begin{bmatrix} 10 & 8 \\ 16 & 7\end{bmatrix} $
Example 2 – Subtracting Matrices on the ACT
Subtract the numbers in the second matrix from those in the same position in the forst matrix, as shown below.
$ \text {Matrix X} \begin{bmatrix} -2 & 9 \\ -4 & 15\end{bmatrix} \text { – Matrix Y} \begin{bmatrix} 6 & 11\\ 3 & -8 \end{bmatrix} $ =
$ \begin{bmatrix} -2 – -6 & 9 – 11\\ -4 – 6 & 11 – -8\end{bmatrix}$ = $ \begin{bmatrix} 4 & -2 \\ -10 & 19\end{bmatrix} $
Example 3 – Multiplying a matrix by a whole number
For these types of ACT matrix problems, you need to multiply each number in the matrix by the whole number:
$ 5 \times \begin{bmatrix} 2 & 4 \\ 8 & 6 \end{bmatrix}$ = $ \begin{bmatrix} 2 \times 5 & 4 \times 5 \\ 8 \times 5 & 6 \times 5\end{bmatrix} $ = $ \begin{bmatrix} 10 & 20 \\ 40 & 30 \end{bmatrix}$
Example 4 – Finding the Determinant of a Matrix
To answer questions on finding the determinant, cross multiply to get two products. Then subtract the two products together for the determinant.
$ \begin{bmatrix} 3 & 11 \\ 9 & 12\end{bmatrix} $
(3 × 12) – (9 × 11) = 36 – 99 = -63
Example 5 – Multiplying 2 Matrices (Method A)
What is Matrix M × Matrix N?
$ \text {Matrix M} \begin{bmatrix} 4 & 5 \\ 8 & 2 \end{bmatrix} \text {Matrix N} \begin{bmatrix} 3 & 6 \\ 9 & 7 \end{bmatrix} $
STEP 1:
Multiply all of the numbers in the first row of Matrix M by the “matching members” in the first column of Matrix N. Then sum these products to get the upper-left number in the new matrix.
$ \text {Matrix M} \begin{bmatrix} 4 & 5 \\ 8 & 2 \end{bmatrix} \text {Matrix N} \begin{bmatrix} 3 & 6 \\ 9 & 7 \end{bmatrix} $
(4 × 3) + (5 × 9) = 12 + 45 = 57
STEP 2:
Multiply each of the numbers in the first row of Matrix M by the “matching members” in the second column of Matrix N. Then sum these products to get the upper-right number in the new matrix.
$ \text {Matrix M} \begin{bmatrix} 4 & 5 \\ 8 & 2 \end{bmatrix} \text {Matrix N} \begin{bmatrix} 3 & 6 \\ 9 & 7 \end{bmatrix} $
(4 × 6) + (5 × 7) = 24 + 35 = 59
STEP 3:
Multiply all of the numbers in the second row of Matrix M by the “matching members” in the first column of Matrix N. Then sum these products to get the lower-left number in the new matrix.
$ \text {Matrix M} \begin{bmatrix} 4 & 5 \\ 8 & 2 \end{bmatrix} \text {Matrix N} \begin{bmatrix} 3 & 6 \\ 9 & 7 \end{bmatrix} $
(8 × 3) + (2 × 9) = 24 + 18 = 42
STEP 4:
Multiply all of the numbers in the second row of Matrix M by the “matching members” in the second column of Matrix N. Then sum these products to get the lower-right number in the new matrix.
$ \text {Matrix M} \begin{bmatrix} 4 & 5 \\ 8 & 2 \end{bmatrix} \text {Matrix N} \begin{bmatrix} 3 & 6 \\ 9 & 7 \end{bmatrix} $
(8 × 6) + (2 × 7) = 48 + 14 = 62
The numbers from these four steps form the matrix: $ \begin{bmatrix} 57 & 59 \\ 42 & 62 \end{bmatrix}$
Example 5 – Multiplying 2 Matrices (Method B)
Alternatively, you may be able to see the operations more clearly if you perform them in matrix format:
$ \begin{bmatrix} 4 & 5 \\ 8 & 2 \end{bmatrix} \times \begin{bmatrix} 3 & 6 \\ 9 & 7 \end{bmatrix} $ =
$ \begin{bmatrix} (4 \times 3) + (5 \times 9) & (4 \times 6) + (5 \times 7) \\ (8 \times 3) + (2 \times 9) & (8 \times 6) + (2 \times 7) \end{bmatrix} $ =
$ \begin{bmatrix} 12 + 45 & 24 + 35 \\ 24 + 18 & 48 + 14 \end{bmatrix}$ = $ \begin{bmatrix} 57 & 59 \\ 42 & 62 \end{bmatrix}$
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