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COMEDK Matrices — practice questions

Explore 59 free Matrices questions in COMEDK Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Matrices questions and exercises with answers

Showing questions 1–30 of 59.

A matrix has $3$ rows and $4$ columns. Its order is:Read the answer and worked solution →The identity matrix $I_n$ of order $n$ is:Read the answer and worked solution →For any matrix $A$, the property $(A^T)^T$ equals:Read the answer and worked solution →The determinant of the $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is:Read the answer and worked solution →A square matrix $A$ has an inverse $A^{-1}$ if and only if:Read the answer and worked solution →A matrix with all elements zero is called:Read the answer and worked solution →For a 3x3 identity matrix I, det(I) equals:Read the answer and worked solution →Number of rows in matrix A times number of columns in A x B should equal:Read the answer and worked solution →The transpose of a 2x3 matrix is a:Read the answer and worked solution →A square matrix A is symmetric if:Read the answer and worked solution →The element a_23 of a matrix is located at:Read the answer and worked solution →If A = [[1,2],[3,4]], find det(A):Read the answer and worked solution →For A = [[2,0],[0,3]] (diagonal matrix), A² equals:Read the answer and worked solution →If A is a 3×3 matrix with det(A) = 2, then det(2A):Read the answer and worked solution →The inverse of a 2x2 matrix [[a,b],[c,d]] is (when det ≠ 0):Read the answer and worked solution →If A·B = I (identity) and A·C = I, then B equals:Read the answer and worked solution →For an n × n matrix A, A is invertible if and only if:Read the answer and worked solution →If A = [[1,2],[3,4]] and B = [[5,6],[7,8]], find A + B:Read the answer and worked solution →If A is 3×4 and B is 4×2, dimensions of AB:Read the answer and worked solution →If A is 3×3 and det(A) = 5, then det(A⁻¹) equals:Read the answer and worked solution →If A is a 3×3 matrix and det(A) = 4, then det(adj A):Read the answer and worked solution →If A is symmetric and B is skew-symmetric (both n × n), then A + B:Read the answer and worked solution →For 2x2 matrix A with characteristic equation λ² - 5λ + 6 = 0, the eigenvalues are:Read the answer and worked solution →By the Cayley-Hamilton theorem, every square matrix satisfies its own:Read the answer and worked solution →For a 3x3 matrix A, A·(adj A) equals:Read the answer and worked solution →Rank of the zero matrix:Read the answer and worked solution →For an orthogonal matrix Q, Q·Qᵀ equals:Read the answer and worked solution →If A is invertible n × n matrix and B is any n × n matrix, then det(A⁻¹BA):Read the answer and worked solution →Determinant of [[1,2,3],[4,5,6],[7,8,9]]:Read the answer and worked solution →Trace of a matrix A is the sum of its:Read the answer and worked solution →

Worked practice questions

A matrix has $3$ rows and $4$ columns. Its order is:

  1. $4 \times 3$
  2. $3 \times 4$
  3. $12$
  4. $7$
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Correct answer: B. $3 \times 4$

Convention: the order of a matrix is written as **rows × columns**. A matrix with 3 rows and 4 columns has order $3 \times 4$. Don't confuse with the total number of elements (12) or the sum of dimensions (7). Matrix multiplication compatibility depends on order: an $m \times n$ matrix can multiply an $n \times p$ matrix.

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The identity matrix $I_n$ of order $n$ is:

  1. A diagonal matrix with all diagonal elements equal to 1 and others 0
  2. A matrix where rows equal columns
  3. A matrix with all elements equal to 1
  4. A matrix with all elements 0
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Correct answer: A. A diagonal matrix with all diagonal elements equal to 1 and others 0

The $n \times n$ identity $I_n$ has $1$s on the main diagonal and $0$s everywhere else. It plays the role of the multiplicative identity: for any compatible $A$, $A \cdot I = I \cdot A = A$. Option A (all 1s) is the $J$ matrix, sometimes called the all-ones matrix. Option C (all zeros) is the zero matrix, the additive identity.

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For any matrix $A$, the property $(A^T)^T$ equals:

  1. $A^{-1}$
  2. $A$
  3. $-A$
  4. $A \cdot A$
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Correct answer: B. $A$

Transposing twice undoes the operation: $(A^T)^T = A$. Intuitively, swapping rows and columns and then swapping them again returns the original matrix. Other useful transpose rules: $(A + B)^T = A^T + B^T$, $(AB)^T = B^T A^T$ (note the reversal), $(kA)^T = k A^T$ for any scalar $k$.

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