Unlock hundreds more features
Save your Quiz to the Dashboard
View and Export Results
Use AI to Create Quizzes and Analyse Results

OR
Sign inSign in with Facebook
Sign inSign in with Google
Quizzes > Mathematics > Algebra

Test Your Matrix Mastery with Our Free Matrices Quiz

Ready for a Matrix Multiplication Quiz? Dive into this Matrices Practice Test!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art geometric grid and numbers representing matrix multiplication determinants transformations on sky blue background.

Think you're mastered matrices? Our free matrices quiz is designed to test and sharpen your skills with engaging challenges in matrix multiplication, determinants, transformations. Dive into a focused matrix multiplication quiz to conquer tricky products, level up with a matrices practice test, or push your limits in a matrix operations quiz. Ideal for students, aspiring data scientists, or anyone cramming for a linear algebra self study quiz, you'll even apply real scenarios with matrices in the real world before tackling a tough matrix math question. Ready to measure your mastery? Start the challenge now and see if you can ace every problem!

What are the dimensions of a matrix with 3 rows and 4 columns?
3x3
3x4
4x3
4x4
The dimensions of a matrix are given by the number of rows followed by the number of columns. Thus a matrix with 3 rows and 4 columns is of size 3x4. This notation is standard in linear algebra. Matrix dimensions explanation
Which of the following describes the 2x2 identity matrix?
A 2x2 matrix with ones on the main diagonal and zeros elsewhere
A 2x2 matrix with all zeros
A 2x2 matrix with all ones
A 2x2 matrix with ones off the main diagonal
The identity matrix of size 2 has ones on its main diagonal and zeros elsewhere, so it leaves any vector unchanged when multiplied. It serves as the multiplicative identity in matrix algebra. This is consistent for any size identity matrix. Identity matrix
Under what condition can two matrices A and B be added?
They must both be square
They must both be invertible
They must have the same dimensions
They must have the same number of rows only
Matrix addition is defined element-wise, so both matrices must share the same number of rows and columns. This ensures each entry in A has a corresponding entry in B. Other conditions like being square or invertible are not required. Matrix addition
In the scalar multiplication 3 * [[2, -1], [0, 4]], what is the element in the second row, first column?
0
12
6
-3
Scalar multiplication multiplies each entry of the matrix by the scalar. The entry in the second row, first column is originally 0, and multiplying by 3 yields 0. All entries are scaled similarly. Scalar multiplication
For A = [[1,2],[3,4]] and B = [[2,0],[1,2]], what is the entry in the first row, second column of A×B?
6
8
2
4
The entry in row 1, column 2 of the product is computed as 1*0 + 2*2 = 4. Each entry of the product is the dot product of the corresponding row of A with the column of B. Matrix multiplication
What is the formula for the determinant of a 2x2 matrix [[a, b], [c, d]]?
ad - bc
ab - cd
ad + bc
ac - bd
The determinant of a 2×2 matrix [[a,b],[c,d]] is calculated as ad minus bc. This scalar value indicates area scaling and orientation change by the linear map. It's a fundamental property in linear algebra. 2×2 determinant
What is the transpose of the matrix [[1, 2, 3], [4, 5, 6]]?
[[3, 2, 1], [6, 5, 4]]
[[1, 2, 3], [4, 5, 6]]
[[1, 4], [2, 5], [3, 6]]
[[1, 4, 5], [2, 3, 6]]
Transposing a matrix swaps its rows and columns. The original 2×3 matrix becomes a 3×2, with entries at (i,j) moved to (j,i). This operation is common in solving linear equations. Matrix transpose
True or False: A square matrix that equals its own transpose is called symmetric.
True
False
A symmetric matrix is one that satisfies A = A?, meaning its entries are mirrored across the main diagonal. This property is important in many areas, including quadratic forms and eigenvalue problems. Symmetric matrices have real eigenvalues and orthogonal eigenvectors. Symmetric matrix
What is the determinant of the matrix [[1, 2, 3], [0, 4, 5], [1, 0, 6]]?
18
24
20
22
The determinant is computed by expansion: 1*(4*6-5*0) - 2*(0*6-5*1) + 3*(0*0-4*1) = 24 +10 -12 = 22. Each minor and cofactor contributes to the final value. Determinants of 3×3 matrices capture volume scaling. 3×3 determinant
If A is invertible, which expression equals (A?)?¹?
(A?¹)?
A? · A?¹
A?¹
(A?)?¹
The inverse of a transpose equals the transpose of the inverse: (A?)?¹ = (A?¹)?. This follows from properties of matrix inverses and transposes. It is frequently used in solving systems and computing adjoints. Inverse properties
What is the rank of the matrix [[1, 2, 3], [2, 4, 6], [3, 6, 9]]?
3
0
2
1
All rows are scalar multiples of [1,2,3], so there is only one linearly independent row. Therefore the rank is 1. Rank measures the dimension of the column or row space. Dependent rows reduce the rank. Matrix rank
Let T? be scaling by 2 and T? a 90° rotation in ?². What is the matrix for T??T??
[[0, 2], [-2, 0]]
[[0, -1], [1, 0]]
[[2, 0], [0, 2]]
[[0, -2], [2, 0]]
Composition applies T? then T?: multiply the rotation matrix [[0,-1],[1,0]] by the scaling matrix [[2,0],[0,2]] on the left. This yields [[0,-2],[2,0]]. Composition of linear maps corresponds to matrix multiplication. Function composition
Is the matrix [[4, 1], [0, 4]] diagonalizable over the real numbers?
Yes, because it is already upper triangular
No, because it has a repeated eigenvalue with a single eigenvector
Yes, because it has two distinct eigenvalues
No, because it is not invertible
A matrix with a repeated eigenvalue can only be diagonalized if it has two independent eigenvectors. This matrix has eigenvalue 4 of algebraic multiplicity 2 but only one eigenvector direction. Therefore it is not diagonalizable over ?. Diagonalizable matrix
0
{"name":"What are the dimensions of a matrix with 3 rows and 4 columns?", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"What are the dimensions of a matrix with 3 rows and 4 columns?, Which of the following describes the 2x2 identity matrix?, Under what condition can two matrices A and B be added?","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}

Study Outcomes

  1. Understand Matrix Multiplication -

    Learn to carry out matrix multiplication step by step and interpret the results within the matrix multiplication quiz context.

  2. Compute Determinants Accurately -

    Master the calculation of determinants for various matrix sizes to assess properties like singularity and volume scaling.

  3. Analyze Geometric Transformations -

    Apply matrices to represent and analyze linear transformations such as rotations, reflections, and scalings in two-dimensional space.

  4. Assess Matrix Invertibility -

    Evaluate when a matrix is invertible and practice finding inverse matrices to solve inverse-related problems in this matrices practice test.

  5. Solve Linear Systems Efficiently -

    Use row operations and inverse matrices to solve systems of linear equations, reinforcing techniques from the matrix operations quiz.

Cheat Sheet

  1. Dimension Compatibility for Multiplication -

    Matrix multiplication requires the number of columns in the first matrix to match the number of rows in the second, e.g., a 2×3 matrix times a 3×4 matrix yields a 2×4 result. This fundamental rule is a staple of any matrix multiplication quiz, so remember the mnemonic "rows to columns, then the product unfolds" to avoid mismatch errors.

  2. Determinant Calculation and Interpretation -

    The determinant of a 2×2 matrix [a b; c d] is ad - bc, while for a 3×3 you can apply Sarrus' Rule by summing diagonals and subtracting counter-diagonals. Determinants measure scaling factors and orientation flips in transformations, which you'll often encounter in a matrices practice test. Keep the formula ad - bc at your fingertips for quick checks in any matrix operations quiz.

  3. Inverse Matrices and Cramer's Rule -

    If det(A)≠0 for a 2×2 matrix A, its inverse is (1/det(A))×[d - b; - c a], a handy formula to master for solving systems in a linear algebra self study quiz. In higher dimensions, you can use the adjugate method or row-reduction to find A❻¹, ensuring you cross-verify by checking AA❻¹=I. Applying Cramer's Rule ties determinants directly to solution variables, reinforcing the role of invertibility in system-solving.

  4. Eigenvalues and Eigenvectors in Transformations -

    An eigenvalue λ and eigenvector v satisfy Av=λv, indicating how the linear transformation scales v along its direction. Recognizing this in a matrices quiz helps you see how rotations or stretches act on spaces, especially when diagonalizing A simplifies computations. Recall the phrase "treasure vectors get treasure scalars" to link "eigen" (German for "own") with their "own" scaling factors.

  5. Rank, Row Reduction, and System Solutions -

    The rank of a matrix equals its number of pivot positions after row-reduction, dictating the dimension of its column space and solution existence for Ax=b. A full-rank square matrix guarantees a unique solution, a concept frequently tested in a matrix operations quiz to connect linear independence and solvability. Use the Rank-Nullity Theorem (rank+nullity=n) as a quick check on the structure of solution sets.

Make a Quiz (FREE) Copy & Edit This Quiz Create a Quiz with AI

Algebra Quizzes

Powered by: Quiz Maker