42 real Linear Algebra questions from the Engineering Mathematics bank, as asked in Indian campus drives and tech interviews. Every question has a verified answer and an AI-tutor explanation on placd — free to start.
1. What is Rank of a matrix?
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A.a real square matrix whose transpose equals its inverse, so its determinant is plus or minus one and it preserves vector length
B.a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector
C.for any matrix, rank plus nullity equals the number of columns
D.the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
A.Rank of a matrix — a linear system is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix
B.Rank of a matrix — its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular
C.Rank of a matrix — every square matrix satisfies its own characteristic equation, which lets integer powers and the inverse be written as a polynomial in the matrix
D.Rank of a matrix — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
A.Rank-nullity theorem — for any matrix, rank plus nullity equals the number of columns
B.Rank-nullity theorem — a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector
C.Rank-nullity theorem — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
D.Rank-nullity theorem — two matrices related by an invertible change of basis, which therefore share characteristic polynomial, eigenvalues, trace, rank and determinant
14. Which term means: "every square matrix satisfies its own characteristic equation, which lets integer powers and the inverse be written as a polynomial in the matrix"?
A.Cayley-Hamilton theorem — every square matrix satisfies its own characteristic equation, which lets integer powers and the inverse be written as a polynomial in the matrix
B.Cayley-Hamilton theorem — a real square matrix whose transpose equals its inverse, so its determinant is plus or minus one and it preserves vector length
C.Cayley-Hamilton theorem — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
D.Cayley-Hamilton theorem — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
A.Rouche-Capelli consistency condition — the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
B.Rouche-Capelli consistency condition — a linear system is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix
C.Rouche-Capelli consistency condition — a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
D.Rouche-Capelli consistency condition — the determinant of a square matrix equals the product of all its eigenvalues counted with multiplicity
A.Determinant as an eigenvalue product — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
B.Determinant as an eigenvalue product — the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
C.Determinant as an eigenvalue product — a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
D.Determinant as an eigenvalue product — the determinant of a square matrix equals the product of all its eigenvalues counted with multiplicity
A.Diagonalisability criterion — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
B.Diagonalisability criterion — the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
C.Diagonalisability criterion — a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
D.Diagonalisability criterion — a square matrix of order n is diagonalisable exactly when it possesses n linearly independent eigenvectors
A.Real symmetric matrix property — its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular
B.Real symmetric matrix property — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
C.Real symmetric matrix property — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
D.Real symmetric matrix property — a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
A.Skew-symmetric matrix property — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
B.Skew-symmetric matrix property — its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular
C.Skew-symmetric matrix property — for any matrix, rank plus nullity equals the number of columns
D.Skew-symmetric matrix property — a real square matrix whose transpose equals its inverse, so its determinant is plus or minus one and it preserves vector length
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