MAT1503 EXAM PACK 2025 {DETAILED QUESTIONS AND ANSWERS}

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Question 1


Consider the following system of linear equations



x1 + x2 − x3 = 2
3x1 + 2x2 − x3 = 3
−x1 − x2 + 2x3 = −1
Write down the augmented matrix, reduce the augmented matrix to generalized
row echelon form and then determine the solution of system


solution :


Augmented Matrix:


1 1 −−1


3 2 1
1 −1 2


Apply the elimination row operations

−3R1 + R2−−−−−−−− > R2

R1 + R3−−−−−−−− > R3

we get

1 −1 −1


0 1 2
0 0 1

The matrix is in generalized row echolen form so the solution is




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x3 = 1, −x2 + 2x3 =

−3, x1 + x2− x3 = 2

thus x2 = 3 + 2x3 =

5, x1 = −x2 + x3 + 2

= −2



Question 2


Consider system as in Question 1


2.1 Write down matrices A and b

2.2 What are the respective sizes of matrices A, x and b

2.3 Determine the inverse A−1 of matrix A


solution:
2.1



A


b



2.2 Dimension of matrix A:
3×3

Dimension of matrix b:
3×1

Dimension of matrix x:
3×1




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2.3


1 1 − −1 1 0
3 2 10 1 0
1 −1 20 0


−3R1 + R2−−−−− > R2

R1 + R3−−−−−−−− > R3
1 −1 −1−1 0
0 1 23 1
0 11 0



−R2−−−−−−−−−− > R2


1 1 −−1 1 − 0
0 1 23 1 0
0 0 11 0


2R3 + R2−−−−−−−−− > R2
R3 + R1−−−−−−−−−−−− > R1


1 1 02 −0
0 1 05 1 2
0 0 11 0
−R2 + R1−−−−−−−− > R1


1 0 0−3 −
0 1 05 1 2
0 0 11 0

Thus the inverse



A




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