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Find the inverse, if possible, for the following matrices.

(a) (8 -5)

11 23

(b) (5 -7 6)

-11 6 2

2 4 -7

(a) (8 -5)

11 23

(b) (5 -7 6)

-11 6 2

2 4 -7

Q2. If A= (1 2 6),

4 11 7

9 13 3

(a) Find the minors of 1,2 and 6.

(b) Find the cofactors of 1,2 and 6.

(c) Evaluate |A|.

(d) A^-1

4 11 7

9 13 3

(a) Find the minors of 1,2 and 6.

(b) Find the cofactors of 1,2 and 6.

(c) Evaluate |A|.

(d) A^-1

1. a) Show that the eigenvalues of a hermitian matrix are real.

Apply Cramer's rule to solve the equation.

2X + Y + Z =4

X - Y +2Z =2

3X - 2Y - Z =0

2X + Y + Z =4

X - Y +2Z =2

3X - 2Y - Z =0

How to obtain the eigenvalues and eigenvectors of the matrix: M=

[2 3 0

3 2 0

0 0 1]

[2 3 0

3 2 0

0 0 1]

Check whether the following system of equations has a solution. 4x+2y+8z+6z=3 2x+2y+2z+2w=1 x+3z+2w=3?

Obtain the eigenvalues and eigenvectors of the matrix:

M= [2 3 0]

[3 2 0]

[0 0 1]

M= [2 3 0]

[3 2 0]

[0 0 1]

Show for a square matrix, the followings are equivalent.

The columns of A are all vectors of length 1, and are all at right angles to each other.

A^T = A^−1

The columns of A are all vectors of length 1, and are all at right angles to each other.

A^T = A^−1

Check whether or not the matrix A=[1 1 1

0 - 2 2

0 - 2 - 3] is diagonalisable. If it is, find a matrix P, and a matrix D such that P^-1 AP=D. If A is not diagonalisable find AdjkA).

0 - 2 2

0 - 2 - 3] is diagonalisable. If it is, find a matrix P, and a matrix D such that P^-1 AP=D. If A is not diagonalisable find AdjkA).

Solve the following equations using matrix algebra:

2x + y - z = 11

x - 2y + 2z = -2

3x - y + 3z = 5

2x + y - z = 11

x - 2y + 2z = -2

3x - y + 3z = 5