Algebra 2 Practice Test: Matrices
- Matrices
- 10 questions
- 15 minutes
These 10 questions practice matrix addition, scalar multiplication, matrix multiplication, dimensions, 2 × 2 determinants and inverses, and matrix equations. Matrices are shown as grids of numbers. To multiply matrices, combine each row of the first with each column of the second.
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Question 1
A =
2 −1 3 4 B =
0 5 −2 1 Which matrix is A + B?
Correct answer: D
Add matching entries: 2 + 0 = 2, −1 + 5 = 4, 3 + (−2) = 1, 4 + 1 = 5. The grid with −6 and 5 comes from subtracting instead of adding.
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Question 2
A =
2 −1 3 4 B =
0 5 −2 1 Which matrix is 3A?
Correct answer: B
Scalar multiplication multiplies every entry by 3: 6, −3, 9, 12. Adding 3 to each entry or multiplying only the first entry are the usual mistakes.
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Question 3
A =
2 −1 3 4 B =
0 5 −2 1 What is the determinant of A?
Correct answer: C
For a 2 × 2 matrix, det = ad − bc = 2 · 4 − (−1) · 3 = 8 + 3 = 11. Ignoring the sign of −1 gives 8 − 3 = 5.
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Question 4
A =
2 −1 3 4 B =
0 5 −2 1 Which matrix is the product AB?
Correct answer: D
Row 1 of A times the columns of B: 2 · 0 + (−1)(−2) = 2 and 2 · 5 + (−1) · 1 = 9. Row 2: 3 · 0 + 4(−2) = −8 and 3 · 5 + 4 · 1 = 19. Multiplying matching entries gives the wrong grid with 0 and −5, and the grid with 15 and 20 is BA.
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Question 5
Matrix P has 2 rows and 3 columns, and matrix Q has 3 rows and 4 columns. What are the dimensions of the product PQ?
Correct answer: A
The inner dimensions (3 and 3) match, so PQ is defined, and its size comes from the outer dimensions: 2 rows and 4 columns.
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Question 6
C =
3 1 5 2 Which matrix is the inverse of C?
Correct answer: C
det C = 3 · 2 − 1 · 5 = 1. Swap the main diagonal entries, change the signs of the other two and divide by the determinant: the inverse has rows 2, −1 and −5, 3. Forgetting to swap 3 and 2 gives the grid that starts with 3.
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Question 7Which matrix has no inverse?
Correct answer: A
A 2 × 2 matrix has no inverse when its determinant is 0. For the grid with rows 2, 6 and 1, 3: 2 · 3 − 6 · 1 = 0. The other determinants are −16, −2 and 4.
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Question 8
D =
4 0 −2 7 1 5 −3 8 6 What is the entry d23?
Correct answer: B
The first subscript is the row and the second is the column: row 2, column 3 is 5. Reading column 2, row 3 gives 8.
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Question 9
The system
x + y = 7
x − y = 1
is written as a matrix equation with coefficient matrix
1 1 1 −1 What is the solution (x, y)?
Correct answer: B
Adding the two equations gives 2x = 8, so x = 4, and then y = 3. Check: 4 + 3 = 7 and 4 − 3 = 1. (3, 4) fails the second equation.
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Question 10
E =
x 3 2 5 The determinant of E is 4. What is x?
Correct answer: A
det E = 5x − 3 · 2 = 5x − 6. Setting 5x − 6 = 4 gives 5x = 10 and x = 2. Forgetting the −6 gives 0.4.
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