Linear Algebra Answers

Show that if A is a matrix with a row of zeros (or a column of zeros) then A is not invertible

 Prove that if A and B are matrices such that A is symmetric, then (BA−1 ) T (A−1BT ) −1 = In.

Suppose V1,V2,VM is linearly independent in V and W€V.Prove that dim span(V1+W1,V2+W,.....VM+W)> or equal to m-1.

2.Suppose U1,U2,.....Um are finite dimensional subspaces of V.Prove that U1+U2+.....Um is finite dimensional and dim(U1+U2+.....Um)<or equal to dim U1+dim U2+.......dim Um.

Prove that if A, B, and C are n × n non-singular matrices, then (ABC)^-1 = C^−1B^−1A^−1 .

Two matrices A and B are equal if

(a) both are rectangular

(b) both have same order

(c) number of columns of A is equal to columns of B

(d) both have same order and equal corresponding elements

choose the correct answer from the multiple choice provided

Find the eigenvalues and eigenvectors of the matrices

1) [ 9 3 ]

2 9

2) 2 0 1

[ 0 2 0 ]

1 0 2

Reduce the quadratic form to a canonical form and find its nature

1) 2xy+2yz+2zx

Assume that T is an n × n matrix with a row of zeros. Prove that T has no inverse.

Use Cramer’s rule to solve for y without solving for x, z and w in the system

2w + x + y + z = 3

−8w − 7x − 3y + 5z = −3

w + 4x + y + z = 6

w + 3x + 7y − z = 1

Give an example of 2 × 2 matrix with non-zero entries that has no inverse.