What condition defines a symmetric matrix?
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.
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What must be true of the diagonal entries of a skew-symmetric matrix?
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.
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Which property characterises an orthogonal matrix?
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.
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What is the determinant of any orthogonal matrix?
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.
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A matrix satisfying the relation that its square equals itself is described as
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.
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A matrix whose square is the identity is described as
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.
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What makes a matrix singular?
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.
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Why is matrix multiplication described as non-commutative?
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.
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What is the transpose of a product AB?
Answer: A
The order reverses, which follows from writing out the entries of the product and comparing them with the transposed factors. The same reversal governs the inverse of a product, and forgetting it is a frequent source of error in manipulation.
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What is the inverse of a product AB, assuming both factors are invertible?
Answer: B
Reversing the order is what allows the product with AB to collapse to the identity, since the inner factors cancel in sequence. The rule mirrors the transpose rule exactly, and both follow from the associativity of the multiplication.
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