Maths questions need help

Question1

A 2X2 matrix A is given by,

a b

c d

Given that a = 9, b = -1, c = 4 and d = 5, determine the entry in the first row, second column of the inverse A-1. Give your answer to 3 decimal places.

Answer

Question 2  

A 2X2 matrix M is given by

a b

0 d

Write down the larger eigenvalue of the matrix 40M where a =2, b=6 and d=15.

Answer

Question 3

A 3X3 matrix is found to have 8 as one of its eigenvalues. It is required to find its eigenvector v = (x1, x2, x3)T. 
The corresponding eigenvector equations are
-1.2×2 – x3 = 0,
2×1 + -1.2×2 = 0, and
2×1 + x3 = 0.
One possible eigenvector was found to be (1, k, -2)T

What is the value of k? Give your answer to 3 decimal places.

Answer

Question 4

If a 3X3 matrix M has characteristic equation 1λ3 + -21λ2 + 5λ = 0.
What is the value of the larger non-zero eigenvalue of M? It is not required to know M exactly.Give your answer to 3 decimal places.

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Question 5

M is a 2X2 singular matrix with tr M = 106. What is the value of the smaller eigenvalue of M? It is not necessary to know M exactly.

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Question 6

A 2X2 matrix M is given by
a b
c d

Given that a = 9, b = 27 and c = 32, what is the value of d such that the matrix M is singular.
Give your answer to the nearest integer.

Answer

Question 7

A system of simultaneous equations in terms of three unknowns, x1, x2 and x3, was solved using Gauss Elimination Method. The final augmented matrix in upper triangular form was given by,

1 a b 8

0 c 3 22

0 0 d 6

When a = -4, b = 4, c = 1 and d = 4, using back substitution, calculate the value of x1, giving your answer to 3 decimal places.

Answer