Matrices and Determinants Questions and Answers

Answer: B

The two matrices commute for all values of a, b, c, d.

Multiplying in both orders:

\[ \begin{aligned} AB &= \begin{bmatrix} ac - bd & ad + bc \\ -(bc + ad) & ac - bd \end{bmatrix} \\ BA &= \begin{bmatrix} ca - db & cb + da \\ -(da + cb) & ca - db \end{bmatrix} \end{aligned} \]

The entries are equal because ordinary numbers commute, whether real or complex. Both matrices have the form \(aI + bJ\) with the same \(J = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\), and any two polynomials in the same matrix commute; such matrices represent complex numbers.

So the restriction to real numbers is unnecessary, and they do not anticommute, since \(AB + BA = 2AB\) is not zero in general.

Answer: B

1

For a triangular matrix the eigenvalues are its diagonal entries: \[ \det(A-\lambda I) = (1-\lambda)^{2} = 0 \] \[ \lambda = 1 \] The eigenvalue is 1, repeated twice, so the options 0, 2 and 3 are not roots.

The other options, 0; 2; 3, are not the answer to this question. This agrees with the official final answer key for the paper, which gives option B.

Answer: A

Equality with the transpose means the entry in row i column j matches that in row j column i, so the matrix is a mirror image across the leading diagonal. Every real symmetric matrix has real eigenvalues and can be diagonalised by an orthogonal matrix, which is why the form matters so much in practice.

Answer: B

Since the transpose is the negative of the matrix, each diagonal entry must equal its own negative, and only zero satisfies that. It follows that the trace of a skew-symmetric matrix is necessarily zero, whatever its off-diagonal entries may be.

Answer: C

The condition means the rows form an orthonormal set, so the matrix preserves lengths and angles under the transformation it represents. Its determinant is therefore plus or minus one, corresponding to a rotation or a rotation with a reflection.

Answer: D

Taking determinants of the defining relation gives the square of the determinant equal to one, so the value is plus or minus one. The positive case is a pure rotation and the negative case a rotation combined with a reflection.

Answer: A

Projection matrices are the standard examples, since projecting twice achieves no more than projecting once. The eigenvalues of such a matrix can only be zero or one, which follows directly from the defining relation applied to an eigenvector.

Answer: B

Such a matrix is its own inverse, which makes reflections the familiar example. Its eigenvalues must satisfy the same relation and so can only be plus or minus one, and its determinant is therefore also plus or minus one.

Answer: C

A zero determinant means no inverse exists, since the inverse formula divides by it, and the rows are linearly dependent. The associated homogeneous system then has non-trivial solutions, which is the practical consequence in solving equations.

Answer: D

The number of columns of the first must match the rows of the second, so one product may exist while the other does not, and even for square matrices the two results usually differ. It is the single most important respect in which matrix algebra departs from ordinary arithmetic.