Matrix | Tensor Analysis Answers

Solve the system of simultaneous equations below using any of the matrix method
2x + 3y - z = 6
3x + y + 5z =8
x + 2y + z =10

4x + 2y + 3z. = 35
x + 3y + 2z. = 45
2x + y + 5z. = 28

Calculat value of x,y & z
Crsmer's rule.

x + y + 2z =5
2 x-3y+5z =5
x + 2y- z. =6
Solve by matrix or adjestment metrix
And also calculate x,y & z.

Q: Solve the following system of equations
3x+2y+4z=7
2x+ y+ z=7
x+3y+5z=2
using Gauss elimination with pivoting. Store the multipliers and also write the pivoting vector.

1. Use technology to solve the appropriate matrix equation to find the quadratic equation that is the
best fit to the points (-2,25), (3,0) , (5,10) and (6,33).
2. Often data is expected to follow an exponential growth model of the form y = A^ekt, where t measures
time and k is called the growth rate. By rewriting the equation as log y = k^t+logA, use this technique
to find the values of A and k that give the best t of the exponential growth model to experimental
data where the values of y at times 0, 1, 2 and 3 are 11, 23, 42 and 80 respectively.

1. Find all values of a, b, c, and d for which A is skew-symmetric.

A = [[0, 2a-3b+c, 3a-5b+5c],[-2, 0, 5a-8b+6c],[-3, -5, d]]

2. Let R be the 5x5 matrix:

[[-8, 33, 38, 173, -30],[0, 0, -1, -4, 0],[0, 0, -5, -25, 1],[0, 0, 1, 5, 0],[4, -16, -19, -86, 15]]

(a) Using technology, and the characteristic polynomial of R and hence and the eigenvalues.

(b) For each of the eigenvalues, determine (by hand) how many linearly independent eigenvectors can be found.

1) M = [ [-2,1,0], [-11,4,1],[-18,6,1]]
a) solve the equation M [ [x] , [y], [z] ] = 3 [ [x] , [y] , [z] ]
b) solve the equation M [ [x], [y], [z] ] = 4 [ [x] , [y], [z] ]

2) A matrix S has eigenvalues 2, -3 and 5 with corresponding eigenvectors v1 = [ [1] , [-3] , [-2] ] , v2 = [ [-2], [7], [5] ] and v3 = [ [0], [0], [1]] respectively.
a) write down the values of Sv1, Sv2 and Sv3.
b) evaluate S[ [0], [1], [0]]
d) hence or otherwise find matrix S

Find a 2x2 matrix X= (abcd) with real entries such that X^2 +2X = -5I

If A and B are two matrices of same order and rank (A) = rank (B) = n, then rank(A+B)=n , for n>=1 (T/F)

if A = 3 2 and B = a b find a b such that
4 1 3 5

AB = BA. Compute 3 A +5B.