Module 2 Quiz: Matrix Algebra
Quiz Instructions
10 questions on matrix operations, transpose, inverses, and LU factorization.
1
If A is 3x4 and B is 4x2, what is the size of AB?
3 x 2. Inner dimensions (4) match; outer dimensions give the result size.
2
Compute: [ 2 1 ; 0 3 ] * [ 1 0 ; -1 2 ].
(1,1)=2-1=1, (1,2)=0+2=2, (2,1)=0-3=-3, (2,2)=0+6=6. Answer: [ 1 2 ; -3 6 ].
3
True or False: For all square matrices A and B, AB = BA.
False. Matrix multiplication is not commutative in general.
4
Find the transpose of [ 1 2 3 ; 4 5 6 ].
[ 1 4 ; 2 5 ; 3 6 ] (3x2 matrix).
5
If (AB)^T = ?
(AB)^T = B^T A^T. The order reverses.
6
Find the inverse of [ 2 5 ; 1 3 ].
det = 6-5 = 1. A^{-1} = [ 3 -5 ; -1 2 ].
7
True or False: If AB = 0, then A = 0 or B = 0.
False. Two nonzero matrices can multiply to give the zero matrix.
8
What is (AB)^{-1} in terms of A^{-1} and B^{-1}?
(AB)^{-1} = B^{-1} A^{-1}. The order reverses.
9
In LU factorization, what kind of matrix is L? What kind is U?
L is lower triangular (with 1s on the diagonal and elimination multipliers below). U is upper triangular (the result of forward elimination -- REF).
10
To solve Ax = b using A = LU, what two simpler systems do you solve?
First solve Ly = b (forward substitution), then solve Ux = y (back substitution).