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Quiz: Matrices and Matrix Operations

Test your understanding of matrices, matrix operations, and their properties.

1. What is the result of multiplying an \(m \times n\) matrix by an \(n \times p\) matrix?

  1. An \(m \times p\) matrix
  2. An \(n \times n\) matrix
  3. An \(m \times n\) matrix
  4. A scalar value
Show Answer

The correct answer is A. When multiplying an \(m \times n\) matrix by an \(n \times p\) matrix, the result is an \(m \times p\) matrix. The inner dimensions (\(n\)) must match, and the outer dimensions (\(m\) and \(p\)) determine the output size.

Concept Tested: Matrix Multiplication

2. Which property does matrix multiplication NOT have?

  1. Associativity: \((AB)C = A(BC)\)
  2. Distributivity: \(A(B + C) = AB + AC\)
  3. Commutativity: \(AB = BA\)
  4. Identity: \(AI = IA = A\)
Show Answer

The correct answer is C. Matrix multiplication is not commutative in general—\(AB \neq BA\) for most matrices. The order of multiplication matters because each product represents a different sequence of transformations.

Concept Tested: Matrix Multiplication Properties

3. What is the transpose of a matrix?

  1. The matrix with all elements negated
  2. The matrix with rows and columns interchanged
  3. The inverse of the matrix
  4. The matrix multiplied by itself
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The correct answer is B. The transpose of a matrix is obtained by interchanging its rows and columns. If \(A\) is an \(m \times n\) matrix, then \(A^T\) is an \(n \times m\) matrix where \((A^T)_{ij} = A_{ji}\).

Concept Tested: Matrix Transpose

4. A square matrix \(A\) is invertible if and only if:

  1. It is symmetric
  2. Its determinant is non-zero
  3. All diagonal elements are positive
  4. It has more rows than columns
Show Answer

The correct answer is B. A square matrix is invertible (has an inverse) if and only if its determinant is non-zero. A zero determinant indicates the matrix is singular and maps some non-zero vectors to zero, making the transformation irreversible.

Concept Tested: Matrix Inverse

5. What is the identity matrix?

  1. A matrix with all ones
  2. A matrix with all zeros
  3. A square matrix with ones on the diagonal and zeros elsewhere
  4. A matrix that equals its transpose
Show Answer

The correct answer is C. The identity matrix \(I\) is a square matrix with ones on the main diagonal and zeros elsewhere. It is the multiplicative identity for matrices: \(AI = IA = A\) for any compatible matrix \(A\).

Concept Tested: Identity Matrix

6. If \(A\) is a \(3 \times 2\) matrix and \(B\) is a \(2 \times 4\) matrix, what are the dimensions of \(AB\)?

  1. \(2 \times 2\)
  2. \(3 \times 4\)
  3. \(4 \times 3\)
  4. \(3 \times 2\)
Show Answer

The correct answer is B. For matrix multiplication \(AB\) where \(A\) is \(3 \times 2\) and \(B\) is \(2 \times 4\), the inner dimensions (2) match, so multiplication is valid. The result has dimensions from the outer dimensions: \(3 \times 4\).

Concept Tested: Matrix Multiplication

7. A symmetric matrix satisfies which condition?

  1. \(A = -A\)
  2. \(A = A^T\)
  3. \(A = A^{-1}\)
  4. \(A = A^2\)
Show Answer

The correct answer is B. A symmetric matrix equals its own transpose: \(A = A^T\). This means \(A_{ij} = A_{ji}\) for all \(i, j\). Symmetric matrices have real eigenvalues and orthogonal eigenvectors.

Concept Tested: Symmetric Matrix

8. What does the rank of a matrix represent?

  1. The number of rows in the matrix
  2. The number of non-zero elements
  3. The dimension of the column space (number of linearly independent columns)
  4. The determinant value
Show Answer

The correct answer is C. The rank of a matrix is the dimension of its column space, which equals the number of linearly independent columns (or equivalently, rows). Rank indicates the effective dimensionality of the transformation the matrix represents.

Concept Tested: Matrix Rank

9. If \(A^{-1}\) exists, what is \(A \cdot A^{-1}\)?

  1. The zero matrix
  2. The identity matrix
  3. The transpose of \(A\)
  4. \(A\) squared
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The correct answer is B. By definition, \(A \cdot A^{-1} = A^{-1} \cdot A = I\), the identity matrix. The inverse "undoes" the transformation performed by the original matrix.

Concept Tested: Matrix Inverse

10. An orthogonal matrix \(Q\) satisfies:

  1. \(Q^T = Q\)
  2. \(Q^T = Q^{-1}\)
  3. \(Q^2 = I\)
  4. \(Q + Q^T = I\)
Show Answer

The correct answer is B. An orthogonal matrix satisfies \(Q^T = Q^{-1}\), which means \(Q^TQ = QQ^T = I\). Orthogonal matrices preserve lengths and angles, representing rotations and reflections.

Concept Tested: Orthogonal Matrix