Question

For each of the following pairs, find an elementary matrix such that E * A = B.

a) A = [1 0; 2 1], B = [3 0; 4 1]
b) A = [2 1; 3 2], B = [4 3; 5 4]
c) A = [0 -1; 2 3], B = [-1 2; 1 4]
d) A = [1 2; 3 4], B = [2 4; 1 3]

Answer 1

To find an elementary matrix such that E * A = B, perform row operations on A to transform it into B. Each row operation can be represented by an elementary matrix.

Step-by-step explanation:

To find an elementary matrix such that E * A = B, we need to perform row operations on A to transform it into B. Each row operation can be represented by an elementary matrix. The product of these elementary matrices will give us the desired E.

Let's go through each pair of matrices:

a) A = [1 0; 2 1], B = [3 0; 4 1]

To transform A into B, we can perform the following row operation: R2 - 2R1. The elementary matrix corresponding to this operation is [1 0; -2 1]. So, E = [1 0; -2 1].

b) A = [2 1; 3 2], B = [4 3; 5 4]

To transform A into B, we can perform the following row operation: R2 - R1. The elementary matrix corresponding to this operation is [1 0; -1 1]. So, E = [1 0; -1 1].

c) A = [0 -1; 2 3], B = [-1 2; 1 4]

To transform A into B, we can perform the following row operation: R1 <-> R2. The elementary matrix corresponding to this operation is [0 1; 1 0]. So, E = [0 1; 1 0].

d) A = [1 2; 3 4], B = [2 4; 1 3]

To transform A into B, we can perform the following row operation: R1 <-> R2. The elementary matrix corresponding to this operation is [0 1; 1 0]. So, E = [0 1; 1 0].