Welcome to the Matrices & Determinants Mastery Quiz. This assessment is designed to test your understanding of intermediate linear algebra concepts, specifically focusing on the properties of adjoints, inverses, and the behavior of systems of linear equations.
What is covered?
- Properties of the Adjoint (adj A) and Inverse (A⁻¹) of square matrices.
- Determinant rules including scalar multiplication and product properties.
- Solving systems of linear equations using Cramer's Rule and matrix methods.
- Consistency and inconsistency conditions for non-homogeneous and homogeneous systems.
Note: For all questions, assume matrices are square and of order n unless specified otherwise.
Q1. If ##A## is a ##3 \times 3## non-singular matrix such that ##|A| = 5##, what is the value of ##|\text{adj}(A)|##?
The property states that ##|\text{adj}(A)| = |A|^{n-1}##. For a ##3 \times 3## matrix, ##5^{3-1} = 5^2 = 25##.
Q2. For a system of linear equations ##AX = B##, if ##|A| \neq 0##, the system is guaranteed to have:
If the determinant of the coefficient matrix is non-zero, the matrix is invertible, and the system has a unique solution given by ##X = A^{-1}B##.
Q3. Which of the following is the correct property for the inverse of a product of two invertible matrices ##A## and ##B##?
The Reversal Law for inverses states that the inverse of a product is the product of the inverses in reverse order.
Q4. If ##A## is a ##3 \times 3## matrix and ##|A| = 2##, then the value of ##|3A|## is:
For an ##n \times n## matrix, ##|kA| = k^n |A|##. Here, ##3^3 \times 2 = 27 \times 2 = 54##.
Q5. The product of a square matrix ##A## and its adjoint ##\text{adj}(A)## is always equal to:
By definition of the adjoint, ##A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A|I##, where ##I## is the identity matrix.
Q6. A homogeneous system of equations ##AX = 0## has a non-trivial solution if and only if:
A homogeneous system always has the trivial solution (zero). It has non-trivial (non-zero) solutions only if the matrix ##A## is singular, meaning ##|A| = 0##.
Q7. If ##A## is a square matrix of order ##n##, then ##\text{adj}(kA)## is equal to:
The property is ##\text{adj}(kA) = k^{n-1} \text{adj}(A)##. This is derived from the fact that each element of the adjoint is a cofactor (determinant of order ##n-1##).
Q8. If ##A## is an invertible matrix, then the determinant of its inverse ##|A^{-1}|## is:
Since ##A \cdot A^{-1} = I##, taking the determinant of both sides gives ##|A| \cdot |A^{-1}| = |I| = 1##, thus ##|A^{-1}| = 1/|A|##.
Q9. If a square matrix ##A## satisfies the equation ##A^2 - 4A - 5I = O##, then ##A^{-1}## is:
Multiply the equation by ##A^{-1}##: ##A - 4I - 5A^{-1} = 0 \implies 5A^{-1} = A - 4I \implies A^{-1} = \frac{1}{5}(A - 4I)##.
Q10. In Cramer's Rule, if ##D = 0## and at least one of ##D_x, D_y, D_z## is non-zero, the system of equations is:
When the main determinant ##D## is zero but at least one coordinate determinant is non-zero, the system represents parallel planes or configurations with no common intersection.
Q11. If ##A## is a ##3 \times 3## matrix such that ##|A| = 3##, what is the value of ##|\text{adj}(\text{adj}(A))|##?
The formula is ##|\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2}##. For ##n=3##, this is ##3^{(3-1)^2} = 3^4 = 81##.
Q12. If ##A## is an orthogonal matrix, then the value of ##|A|## must be:
For an orthogonal matrix, ##AA^T = I##. Taking determinants: ##|A| \cdot |A^T| = 1##. Since ##|A^T| = |A|##, we have ##|A|^2 = 1##, so ##|A| = \pm 1##.
Q13. Which of the following is true regarding the adjoint of a symmetric matrix?
If ##A## is symmetric (##A = A^T##), its adjoint ##\text{adj}(A)## is also symmetric because the cofactor matrix of a symmetric matrix is symmetric.
Q14. The inverse of a symmetric matrix, if it exists, is:
If ##A = A^T##, then ##(A^{-1})^T = (A^T)^{-1} = A^{-1}##. Thus, the inverse is also symmetric.
Q15. If ##A## and ##B## are two square matrices of the same order such that ##AB = O## and ##A## is non-singular, then:
If ##A## is non-singular, ##A^{-1}## exists. Multiplying ##AB = O## by ##A^{-1}## on the left gives ##A^{-1}AB = A^{-1}O \implies IB = O \implies B = O##.
Q16. If ##A## is a ##2 \times 2## matrix ##[[a, b], [c, d]]##, what is ##\text{adj}(A)##?
For a ##2 \times 2## matrix, the adjoint is found by swapping the main diagonal elements and changing the signs of the off-diagonal elements.
Q17. A system of ##n## linear equations in ##n## variables ##AX = B## has infinitely many solutions if:
If the determinant is zero and the product of the adjoint and the constant matrix results in a zero matrix, the system is consistent with infinite solutions (dependent equations).
Q18. What is ##\text{adj}(I)##, where ##I## is the identity matrix of order ##n##?
Since ##I \cdot \text{adj}(I) = |I|I = 1 \cdot I = I##, it follows that ##\text{adj}(I) = I##.
Q19. If ##A## is a square matrix, then ##(A^T)^{-1}## is equal to:
The operations of taking the transpose and taking the inverse are commutative: the inverse of the transpose is the transpose of the inverse.
Q20. If ##A## is a skew-symmetric matrix of odd order, then ##|A|## is:
The determinant of a skew-symmetric matrix of odd order is always zero. Such matrices are always singular.
Q21. If ##A## is a square matrix such that ##A^2 = A##, then which of the following is true for ##(I + A)^3##?
Expanding ##(I + A)^3 = I^3 + 3I^2A + 3IA^2 + A^3##. Since ##A^2 = A## and ##A^3 = A \cdot A^2 = A \cdot A = A##, it becomes ##I + 3A + 3A + A = I + 7A##.
Q22. The value of the determinant remains unchanged if:
Adding a scalar multiple of one row to another row (Elementary Row Operation Type 3) does not change the value of the determinant.
Q23. If ##A## is a ##3 \times 3## matrix and ##|A| = 4##, find ##|A \cdot \text{adj}(A)|##.
Since ##A \cdot \text{adj}(A) = |A|I##, then ##|A \cdot \text{adj}(A)| = | |A|I | = |A|^n |I| = 4^3 \times 1 = 64##.
Q24. If the system ##x + y + z = 6##, ##x + 2y + 3z = 10##, ##x + 2y + \lambda z = \mu## has infinite solutions, then:
For infinite solutions, the third equation must be a linear combination of the others or identical to the second. If ##\lambda = 3## and ##\mu = 10##, the second and third equations are identical, leading to a dependent system.
Q25. The inverse of a non-singular diagonal matrix ##\text{diag}(d_1, d_2, ..., d_n)## is:
The inverse of a diagonal matrix is simply a diagonal matrix where each diagonal element is replaced by its reciprocal.
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