Chapter 3 True/False Linear Algebra
Adding a multiple of one column of a matrix to another column changes only the sign of the determinant.
False
If A and B are square matrices of order n, then det (A + B) = det(A) + det(B).
False
If A is a 3 X 3 matrix with det(A) = 5, then det (2A) = 10.
False
If A is a square matrix of order n, then det(A) = -det(A^T).
False
If A is an n X n matrix and c is a nonzero scalar, then the determinant of the matrix cA is given by nc X det(A).
False
If a square matrix B is obtained from A by interchanging two rows, then det(B) = det(A).
False
In general, the determinant of the sum of two matrices equals the sum of the determinants of the matrices.
False
The cofactor C22 of a given matrix is always a positive number.
False
The determinant of a 2 X 2 matrix A is a21a12-a11ba22.
False
The ij-cofactor of a square matrix A is the matrix defined by deleting the ith row and jth column of A.
False
To find the determinant of a triangular matrix, add the entries on the main diagonal.
False
If A and B are square matrices of order n such that det(AB) = -1, then both A and B are nonsingular.
True
If A and B are square matrices of order n, and det(A) = det(B), the det(AB) = det(A^2).
True
If A is an invertible matrix, then the determinant of A^-1 is equal to the reciprocal of the determinant of A.
True
If A is an invertible n X n matrix, then Ax = b has a unique solution for every b.
True
If one column of a square matrix is a multiple of another column, then the determinant is O.
True
If one row of a square matrix is a multiple of another row, then the determinant is 0.
True
If the determinant of an n X n matrix A is nonzero, then Ax = O has only the trivial solution.
True
If two rows of a square matrix are equal, then its determinant is 0.
True
Interchanging two rows of a given matrix changes the sign of its determinant.
True
Multiplying a column of a matrix by a nonzero constant results in the determinant being multiplied by the same nonzero constant.
True
The determinant of a matrix of order 1 is the entry of the matrix.
True
To find the determinant of a matrix, expand by cofactors in any row or column.
True
Two matrices are column-equivalent when one matrix can be obtained by performing elementary column operations on the other.
True
When expanding by cofactors, you need not evaluate the cofactors of zero entries.
True
