Question

Use the fact that matrices A and B are row-equivalent.

A = −2 −5 8 0 −17 1 3 −5 1 5 −5 −11 17 3 −53 1 7 −13 5 −3
B = 1 0 1 0 1 0 1 −2 0 3 0 0 0 1 −5 0 0 0 0 0
a) Calculate the determinant of matrix A.
b) Find the inverse of matrix B.
c) Verify if matrix A is symmetric.
d) Find the reduced row-echelon form of matrix B.

Answer 1

The task involves a series of matrix operations: calculating the determinant and inverse, verifying symmetry, and determining the reduced row-echelon form for given matrices A and B.

Step-by-step explanation:

The question pertains to matrix operations and properties, and involves an understanding of row-equivalence of matrices, calculation of determinants, finding matrix inverses, verifying matrix symmetry, and reduced row-echelon forms.

1. To calculate the determinant of matrix A, apply the determinant rules for a 4x4 matrix, which generally involves expansion by minors or using a calculator if allowed.
2. The inverse of matrix B can be found by augmenting matrix B with the identity matrix of the same size and applying row operations to obtain the identity matrix on the left side; the right side of the augmented matrix will then be the inverse of B.
3. To verify if matrix A is symmetric, check if the transpose of matrix A equals matrix A itself.
4. The reduced row-echelon form of matrix B is found by applying row operations until each leading entry is 1 and is the only nonzero entry in its column, and all entries above and below the leading entry are zero.