# MATH 317 MID-TERM EXAM

**Note:
**[1] There should be two parts.

[2] Read the questions carefully before answering.

[3] Show all your work to receive credit.

**PART A (15 points)
**State true or false or fill in the blanks with appropriate answers, for each
of the following

statement (assume all matrix operations are valid).

(1) When both A and B are nonsingular (AB)^{-1} = B^{-1} A^{-1}.

(2) A is nonsingular if and only if det(A) is nonzero.

(3) Rank (AB) is always equal to rank of A.

(4) det(A + B) = det(A) + det(B).

(5) If A and B are square matrices, then det(AB) = det(A)det(B).

(6) When A inverse exists, det(A^{-1}) = 1/det(A).

(7) A square matrix A is said to be skew-symmetric iff

(8) The system Ax = b will have solution iff rank (A) = rank[A b].

(9) It is possible for the system Ax = 0 to be inconsistent.

(10) W = {(x,0)^{T}: x is a real number} is a subspace of R^{2} with standard
operations.

(11) V = {set of all 3 × 3 nonsingular matrices} is a
subspace of the set of all 3 x 3 matrices with

standard operations.

(12) For the

the trace is___ and the determinant is

(13) Suppose that x is a nonzero vector. Then Ax = 0 iff A is singular.

(14) A Adj(A) = det(A) I is always true when and only when A is nonsingular.

(15) It is possible for a p x q matrix A to have neither a left- nor a
right-inverse.

**PART B (85 points)
**[1] Given
and
find

(c) the complex conjugate of

[2] Show that for all complex numbers z and w

[3] Suppose that A and B are p x p matrices such that AB is nonsingular. Show that A and B both

are nonsingular.

[4] Show that B is row-equivalent to A. Show that there exists a nonsingular matrix F such that

FA = B.

[5] Solve the system of linear equations AX = B, where A, X, and B are given by

[6] Show that every Hermitian matrix is the sum of a real symmetric matrix and i times a real

skew-symmetric matrix.

[7] Suppose that A is a nonsingular matrix of order 3, reduced to an identity matrix by

performing the following elementary row operations (in the same order): (

Find the matrix A

^{-1}, by first writing A as a product of elementary matrices.

[8] Suppose that B is p x q and A is a p x p nonsingular matrix. Show that the rank of AB equals

the rank of B.

[9] Given the matrix

find the LU decomposition of A

[10] If A is a square matrix of order p, show that