If a problem is solved. It is not 'the' answer.
No attempt is made to search for the most elegant answer.
I highly recommend that you at least try to solve the
problem before you read the solution.
A and B are n x n matrices. Investigate if (A + B)^{2} = A^{2} + 2.A.B + B^{2} |
We say that two matrices A and B commute, if and only if AB = BA. Show that all matrices of the form [a -b] [b a] commute. |
Prove that for each n x n matrix A , A^{T} .A is symmetric |
Prove that if matrix A is skew-symmetric, then A.A is symmetric. |
Given X = [x y] and A is a 2 x 2 matrix . All elements of the matrices are real numbers. Prove that X . A^{T} . A . X^{T} can't be negative. |
Determine for which values of a and b the following system has no solutions.
/ ax + y + 2z = 0 | x + 2y + z = b \ 2x + y + az = 0 |
Calculate the m-values such that the following system has more than one solution.
/ (m+2)x + 2y + 4z = 3m | -mx + 5y + 2mz = -2m \ 2x + 7y + 6z = 1 |
[ 1 m - 1 2 m - 3 ] Given A = [ m 2 m - 2 2 ] [ m + 1 3 m - 3 m.m - 1 ] Calculate the condition for m such that A is regular. Assume that m satisfies this condition and consider the system with A as coefficient matrix. / x + (m - 1)y + (2 m - 3)z = 1 | | m x + (2 m - 2)y + 2 z = 0 | \ (m + 1)x + (3 m - 3)y + (m.m - 1)z = 0 Now, let x = 1 and calculate the values of m such that the system has a solution for y and z. |
The matrix x is a 2 x 2 matrix. Calculate three solutions of the quadratic equation x^{2} - x = 0 |
[-2 -9 ] n [1-3n -9n ] Given A = [ ] . Prove that A = [ ] [ 1 4 ] [ n 1+3n ] |
[1 1 0 ] Given : A = [0 1 0 ] [0 0 1 ] Show that A is regular. n Calculate A . Calculate the real numbers a and b such that A^{2} + a A + b I = 0 ( I is the 3 x 3 identity matrix) Show that there are real numbers c_{0},c_{1},c_{2}, ... ,c_{n} such that A^{-n} = c_{0}.I + c_{1}.A^{2} + c_{2}.A^{3} + c_{3}.A^{4} + c_{4}.A + ... + c_{n}.A^{n} |