Math 33A Practice Problems IWritten by Victoria Kalavtkala@math.ucla.edu

Last updated April 22, 2019

1. Circle true (T) or false (F) for each statement below. Optional challenge: Can you ex-

plain/prove why each statement is true or false?

(a) T F It is possible for a system of linear equations to have exactly three solutions.

(b) T F The following system is consistent:

x2 � 4x3 = 8

2x1 � 3x2 + 2x3 = 1

4x1 � 8x2 + 12x3 = 1

(c) T F The following system has a unique solution:

x1 � 2x2 + x3 = 0

2x2 � 8x2 = 8

5x1 � 5x3 = 10

(d) T F The vector (3,�1) is a solution to the system Ax = b where

A =

0

@1 2�2 5�5 6

1

A , b =

0

@74�3

1

A

(e) T F Any system with a 3⇥ 3 matrix with rank 3 has a unique solution.

(f) T F Any system with a 3⇥ 5 matrix with rank 3 has a unique solution.

(g) T F Any system with a 4⇥ 3 matrix with rank 3 has a unique solution.

(h) T F For any two matrices A,B, the sum A+B is always defined.

(i) T F All transformations are linear transformations.

(j) T F Every transformation T (x) can be written as T (x) = Ax for some matrix A.

(k) T F Every linear transformation T (x) can be written as T (x) = Ax for some matrixA.

(l) T F The transformation T (x) =

✓0 00 1

◆x projects x = (x, y) onto the y-axis.

(m) T F If A and B are square matrices, then AB = BA.

(n) T F For any two matrices A,B, the product AB is always defined.

(o) T F All square matrices have an inverse.

(p) T F If A and B are invertible matrices then (AB)�1 = A�1B�1.

(q) T F If A and B are invertible matrices then (AB)�1 = B�1A�1.

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