Linear Algebra interview questions

1. A.a real square matrix whose transpose equals its inverse, so its determinant is plus or minus one and it preserves vector length
2. B.a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector
3. C.for any matrix, rank plus nullity equals the number of columns
4. D.the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor

Answer + AI explanation with Pro

1. A.Cayley-Hamilton theorem
2. B.Rank of a matrix
3. C.Diagonalisability criterion
4. D.Similar matrices

Answer + AI explanation with Pro

1. 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
2. B.Rank of a matrix — its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular
3. 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
4. D.Rank of a matrix — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor

Answer + AI explanation with Pro

1. A.the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
2. B.a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
3. C.for any matrix, rank plus nullity equals the number of columns
4. D.a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector

Answer + AI explanation with Pro

1. A.Rank of a matrix
2. B.Diagonalisability criterion
3. C.Real symmetric matrix property
4. D.Rank-nullity theorem

Answer + AI explanation with Pro

1. A.Rank-nullity theorem — for any matrix, rank plus nullity equals the number of columns
2. B.Rank-nullity theorem — a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector
3. C.Rank-nullity theorem — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
4. D.Rank-nullity theorem — two matrices related by an invertible change of basis, which therefore share characteristic polynomial, eigenvalues, trace, rank and determinant

Answer + AI explanation with Pro

1. A.a square matrix of order n is diagonalisable exactly when it possesses n linearly independent eigenvectors
2. B.a real square matrix whose transpose equals its inverse, so its determinant is plus or minus one and it preserves vector length
3. C.for any matrix, rank plus nullity equals the number of columns
4. D.a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector

Answer + AI explanation with Pro

1. A.Trace of a matrix
2. B.Eigenvalue
3. C.Skew-symmetric matrix property
4. D.Diagonalisability criterion

Answer + AI explanation with Pro

1. A.Eigenvalue — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
2. B.Eigenvalue — a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector
3. C.Eigenvalue — a real square matrix whose transpose equals its inverse, so its determinant is plus or minus one and it preserves vector length
4. D.Eigenvalue — a square matrix of order n is diagonalisable exactly when it possesses n linearly independent eigenvectors

Answer + AI explanation with Pro

1. A.its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
2. B.the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
3. C.the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
4. D.its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular

Answer + AI explanation with Pro

1. A.Rouche-Capelli consistency condition
2. B.Real symmetric matrix property
3. C.Skew-symmetric matrix property
4. D.Trace of a matrix

Answer + AI explanation with Pro

1. A.Trace of a matrix — a square matrix of order n is diagonalisable exactly when it possesses n linearly independent eigenvectors
2. B.Trace of a matrix — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
3. C.Trace of a matrix — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
4. D.Trace of a matrix — the sum of the main-diagonal entries, which also equals the sum of the eigenvalues

Answer + AI explanation with Pro

1. A.a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector
2. B.a linear system is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix
3. C.the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
4. D.every square matrix satisfies its own characteristic equation, which lets integer powers and the inverse be written as a polynomial in the matrix

Answer + AI explanation with Pro

1. A.Trace of a matrix
2. B.Positive definite matrix
3. C.Skew-symmetric matrix property
4. D.Cayley-Hamilton theorem

Answer + AI explanation with Pro

1. 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
2. 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
3. C.Cayley-Hamilton theorem — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
4. D.Cayley-Hamilton theorem — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor

Answer + AI explanation with Pro

1. A.a linear system is consistent exactly when the rank of the coefficient matrix equals the rank of the augmented matrix
2. B.a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector
3. C.the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
4. D.a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive

Answer + AI explanation with Pro

1. A.Rouche-Capelli consistency condition
2. B.Rank-nullity theorem
3. C.Similar matrices
4. D.Trace of a matrix

Answer + AI explanation with Pro

1. A.Rouche-Capelli consistency condition — the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
2. 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
3. C.Rouche-Capelli consistency condition — a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
4. D.Rouche-Capelli consistency condition — the determinant of a square matrix equals the product of all its eigenvalues counted with multiplicity

Answer + AI explanation with Pro

1. A.for any matrix, rank plus nullity equals the number of columns
2. B.its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular
3. C.the determinant of a square matrix equals the product of all its eigenvalues counted with multiplicity
4. D.the sum of the main-diagonal entries, which also equals the sum of the eigenvalues

Answer + AI explanation with Pro

1. A.Skew-symmetric matrix property
2. B.Determinant as an eigenvalue product
3. C.Orthogonal matrix
4. D.Positive definite matrix

Answer + AI explanation with Pro

1. A.Determinant as an eigenvalue product — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
2. B.Determinant as an eigenvalue product — the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
3. C.Determinant as an eigenvalue product — a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
4. D.Determinant as an eigenvalue product — the determinant of a square matrix equals the product of all its eigenvalues counted with multiplicity

Answer + AI explanation with Pro

1. A.the determinant of a square matrix equals the product of all its eigenvalues counted with multiplicity
2. B.the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
3. C.a square matrix of order n is diagonalisable exactly when it possesses n linearly independent eigenvectors
4. D.its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular

Answer + AI explanation with Pro

1. A.Eigenvalue
2. B.Similar matrices
3. C.Skew-symmetric matrix property
4. D.Diagonalisability criterion

Answer + AI explanation with Pro

1. A.Diagonalisability criterion — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
2. B.Diagonalisability criterion — the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
3. C.Diagonalisability criterion — a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
4. D.Diagonalisability criterion — a square matrix of order n is diagonalisable exactly when it possesses n linearly independent eigenvectors

Answer + AI explanation with Pro

1. A.a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive
2. B.the sum of the main-diagonal entries, which also equals the sum of the eigenvalues
3. C.a square matrix of order n is diagonalisable exactly when it possesses n linearly independent eigenvectors
4. D.its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal

Answer + AI explanation with Pro

1. A.Orthogonal matrix
2. B.Diagonalisability criterion
3. C.Determinant as an eigenvalue product
4. D.Real symmetric matrix property

Answer + AI explanation with Pro

1. A.Real symmetric matrix property — its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular
2. B.Real symmetric matrix property — its eigenvalues are always real and eigenvectors belonging to distinct eigenvalues are mutually orthogonal
3. C.Real symmetric matrix property — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
4. D.Real symmetric matrix property — a symmetric matrix all of whose eigenvalues are strictly positive, equivalently all leading principal minors are strictly positive

Answer + AI explanation with Pro

1. A.its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular
2. B.a scalar for which the matrix equation Ax equals that scalar times x admits a non-zero solution vector
3. C.every square matrix satisfies its own characteristic equation, which lets integer powers and the inverse be written as a polynomial in the matrix
4. D.the sum of the main-diagonal entries, which also equals the sum of the eigenvalues

Answer + AI explanation with Pro

1. A.Eigenvalue
2. B.Rank-nullity theorem
3. C.Real symmetric matrix property
4. D.Skew-symmetric matrix property

Answer + AI explanation with Pro

1. A.Skew-symmetric matrix property — the number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor
2. B.Skew-symmetric matrix property — its eigenvalues are either zero or purely imaginary, and every odd-order such matrix is singular
3. C.Skew-symmetric matrix property — for any matrix, rank plus nullity equals the number of columns
4. 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

Answer + AI explanation with Pro