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Quizzes > Quizzes for Business > Education

Matrix and Determinant Knowledge Test Quiz

Challenge Your Matrix and Determinant Skills

Difficulty: Moderate
Questions: 20
Learning OutcomesStudy Material
Colorful paper art displaying questions for Matrix and Determinant Knowledge Test

Ready to test your grasp of matrix transformations and determinant calculations? This matrix quiz is perfect for students, educators, and programming enthusiasts looking to deepen their understanding of linear algebra fundamentals. Alongside technical C/C++ Fundamentals Knowledge Test and insightful IT Fundamentals Knowledge Test, it offers a hands-on way to reinforce concepts. You can customize every question in our editor to fit your learning goals. Explore more quizzes to continue your skill-building journey.

What is the determinant of the 2� - 2 matrix [[2, 3], [1, 4]]?
5
-5
4
10
The determinant of a 2� - 2 matrix [[a,b],[c,d]] is ad �' bc; here 2*4 �' 3*1 = 8 �' 3 = 5. This formula subtracts the product of off-diagonal entries from the product of diagonal entries. Hence, the determinant is 5.
If A is the 3� - 3 identity matrix, what is det(A)?
-1
0
1
3
The identity matrix has ones on the diagonal and zeros elsewhere, so its determinant is the product of its diagonal entries: 1 � - 1 � - 1 = 1. The determinant measures volume scaling, and identity leaves volumes unchanged, giving a determinant of 1.
Which elementary row operation changes the sign of a determinant?
Adding a multiple of one row to another
Multiplying a row by a scalar
Transposing the matrix
Swapping two rows
Swapping two rows multiplies the determinant by �'1, so it changes the sign of the determinant. Other row operations either leave the determinant unchanged or scale it by a nonnegative factor.
What is the transpose of the matrix [[1,2],[3,4]]?
[[1,3],[2,4]]
[[1,2],[3,4]]
[[3,4],[1,2]]
[[2,1],[4,3]]
The transpose of a matrix flips its rows and columns, so the element at position (i,j) moves to position (j,i). Transposing [[1,2],[3,4]] yields [[1,3],[2,4]].
Which of the following 2� - 2 matrices is singular?
[[2,3],[1,4]]
[[1,0],[0,1]]
[[1,2],[2,4]]
[[0,1],[1,0]]
The determinant of [[1,2],[2,4]] is 1*4 �' 2*2 = 0, indicating the matrix is singular. A singular matrix has a zero determinant and is not invertible.
What is the determinant of the upper triangular matrix [[1,2,3],[0,4,5],[0,0,6]]?
48
24
120
6
For a triangular matrix, the determinant equals the product of the diagonal entries: 1� - 4� - 6 = 24. Off-diagonal entries do not affect this product in triangular form.
If you add three times the first row to the second row of a matrix, how does the determinant change?
It triples
It remains the same
It changes sign
It doubles
Adding a multiple of one row to another is a shear operation that leaves the determinant unchanged. Only row swaps or scaling operations alter the determinant's value.
Multiplying a single row of an n� - n matrix by scalar k multiplies the determinant by what factor?
1/k
-k
k
k^n
Scaling a single row by k multiplies the determinant by k due to the multilinear property of determinants. No other factors change when only one row is scaled.
Compute the determinant of the matrix [[2,0,1],[3,0,0],[5,1,1]].
3
7
-3
5
By expanding along the second column or by direct calculation, one finds the determinant equals 3. The zero entries simplify the computation to a small minor.
In the system x + 2y = 5 and 3x - y = 4, what is the value of x using Cramer's rule?
1
13/7
-13/7
5/7
The determinant of the coefficient matrix is (1 * -1) - (2 * 3) = -7, and the x-determinant is (5 * -1) - (2 * 4) = -13. Using Cramer's rule x = (-13)/(-7) = 13/7.
If det(A) = 3 and det(B) = -2 for two 3� - 3 matrices A and B, what is det(2A · B^T)?
-48
48
-24
24
Since det(2A) = 2^3 � - det(A) = 8 � - 3 = 24 for a 3� - 3 matrix and det(B^T) = det(B) = -2, their product is 24 � - (-2) = -48. The determinant of a transpose equals the determinant of the original matrix.
What is the determinant of the matrix [[1,2,3],[4,5,6],[7,8,9]]?
0
-3
15
45
The rows of the matrix are linearly dependent (the third row equals the sum of the first two), so the determinant is zero. Direct expansion of minors also confirms a result of zero.
Which of the following is a true determinant property for square matrices A and B of the same size?
det(AB) = det(A) + det(B)
det(A^T) = -det(A)
det(AB) = det(A)det(B)
det(A + B) = det(A)det(B)
The determinant of a product of two matrices equals the product of their determinants. This fundamental property holds for any pair of square matrices of the same dimensions.
If det(A) = d � 0 for an invertible n� - n matrix A, what is det(A^{-1})?
d^(n-1)
d
-d
1/d
The product of a matrix and its inverse is the identity, whose determinant is 1, so det(A^{-1}) = 1/det(A). The inverse thus inverts the scaling effect of the original matrix.
In the system x + 2y = 5 and 3x - y = 4, what is the value of y using Cramer's rule?
11/7
-11/7
5/7
2
The coefficient determinant is (1 * -1) - (2 * 3) = -7, and the y-determinant is (1 * 4) - (5 * 3) = -11. By Cramer's rule, y = (-11)/(-7) = 11/7.
Compute the determinant of the 4� - 4 matrix [[2,1,0,0],[3,4,1,0],[0,2,5,3],[0,0,1,6]].
120
111
90
105
Expanding along the first column gives 2 � - det([[4,1,0],[2,5,3],[0,1,6]]) �' 3 � - det([[1,0,0],[2,5,3],[0,1,6]]). Evaluating these minors yields 2� - 96 �' 3� - 27 = 192 �' 81 = 111.
Suppose det(A) = 7. After swapping two rows in A, multiplying one row by 3, and then adding five times one row to another, what is det of the resulting matrix?
-7
7
21
-21
Swapping two rows multiplies the determinant by -1, multiplying a row by 3 scales the determinant by 3, and adding a multiple of one row to another leaves it unchanged. Therefore the new determinant is 7 � - (-1) � - 3 = -21.
In the system x + y + z = 6 2x - y + z = 3 x + 2y - z = 2 what is x using Cramer's rule?
1
3
0
2
The coefficient matrix determinant is 7 and the determinant replacing the x-column with constants is also 7. Therefore x = 7/7 = 1 by Cramer's rule.
For an n� - n matrix A with adjugate adj(A), what is det(adj(A)) in terms of det(A)?
det(A)^(n-2)
det(A)^n
1/det(A)
det(A)^(n-1)
The adjugate of A satisfies adj(A) = det(A)A^{-1}, so det(adj(A)) = det(det(A)A^{-1}) = det(A)^{n-1}. This uses the property that scaling a matrix by a factor scales its determinant by that factor raised to the matrix size.
If a 3� - 3 matrix A has eigenvalues 2, 3, and 5, what is det(A^{-1})?
10
1/10
1/30
30
The determinant of A is the product of its eigenvalues: 2 � - 3 � - 5 = 30. Hence det(A^{-1}) = 1/det(A) = 1/30.
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Learning Outcomes

  1. Analyse matrix operations and properties
  2. Evaluate determinant values of square matrices
  3. Apply row operations to simplify matrices
  4. Demonstrate understanding of determinant theorems
  5. Identify singular and invertible matrices by determinants
  6. Solve systems using Cramer's rule

Cheat Sheet

  1. Master Basic Matrix Operations - Get cozy with adding, subtracting, and multiplying matrices, and discover why AB isn't the same as BA. Practice these rules with fun examples to see how distributive and associative properties play out in action. MIT OCW: Basic Matrix Operations
  2. Calculate Determinants of Square Matrices - Learn the recipe for finding determinants of 2×2 and 3×3 matrices using cofactor expansion, from ad−bc to more elaborate minors. These calculations are your secret weapon for solving systems and testing invertibility. MIT OCW: Determinants Deep Dive
  3. Apply Row Operations to Simplify Matrices - Swap, scale, and combine rows like a pro to simplify matrices for Gaussian elimination. Just remember: each tweak can change the determinant, so track those sign flips and scalar factors! MIT OCW: Row Operations
  4. Understand Determinant Theorems - Unlock shortcuts by exploring theorems like det(AB)=det(A)×det(B) and how scaling a row stretches volumes. These nifty rules let you break big problems into bite-sized pieces. MIT OCW: Determinant Theorems
  5. Identify Singular and Invertible Matrices - Spot invertible matrices by checking for a nonzero determinant; hit zero and you're in singular territory - no inverse coming! This concept is crucial for understanding when linear systems have unique solutions. MIT OCW: Invertibility Essentials
  6. Solve Systems Using Cramer's Rule - Turn determinants into solutions by replacing columns and dividing by the main determinant - perfect for 2×2 and 3×3 systems. It's a neat trick that shines in small puzzles but gets heavy for big ones. MIT OCW: Cramer's Rule
  7. Explore the Adjugate and Inverse of a Matrix - Dive into the formula A❻¹=(1/det(A))×adj(A) and see how cofactors and transposes team up to build the adjugate. It's a perfect mix of determinant wisdom and matrix mechanics. MIT OCW: Adjugate & Inverse
  8. Understand the Geometric Interpretation of Determinants - Think of determinants as volume multipliers: the absolute value gives the volume of the parallelepiped spanned by column vectors. This bridges algebraic skills with visual intuition. MIT OCW: Geometric View
  9. Recognize Limitations of Cramer's Rule - While elegant for small systems, Cramer's Rule demands lots of determinants for larger matrices, making Gaussian elimination a speedier alternative. Choose your tool wisely! MIT OCW: Rule vs. Elimination
  10. Practice with Real-World Applications - Apply these concepts to electrical circuits, economic models, or 3D graphics transformations to see linear algebra in action. Real problems make theory stick like glue! MIT OCW: Real-World Uses
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