Page 331 - 35Linear Algebra

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Sample Second Midterm

Here are some worked problems typical for what you might expect on a second
midterm examination.

a b
1. Determinants: The determinant det M of a 2 × 2 matrix M = is
c d
defined by
det M = ad − bc .

(a) For which values of det M does M have an inverse?
(b) Write down all 2×2 bit matrices with determinant 1. (Remember bits
are either 0 or 1 and 1 + 1 = 0.)
(c) Write down all 2 × 2 bit matrices with determinant 0.
(d) Use one of the above examples to show why the following statement is
FALSE.
Square matrices with the same determinant are always row
equivalent.

2. Let
 
1 1 1
 
A =  2 2 3  .
 
4 5 6
 
1
Compute det A. Find all solutions to (i) AX = 0 and (ii) AX =  2  for
3

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