Exercise miscellaneous

If A and B are symmetric matrices, prove that AB - BA is a skew symmetric matrix.

Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

A / y z

   = −    −  satisfy the equation A′A = I.

For what values of x : [ ] 1 2 0 0 1 2 1 2 0 1 2 1 0 2 x                    = O?

If 3 1A 1 2  =  −  , show that A2 - 5A + 7I = 0.

Find x, if [ ] 1 0 2 5 1 0 2 1 4 O 2 0 3 1 x x        − − =          

A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below: Market Products I 10,000 2,000 18,000 II 6,000 20,000 8,000 (a) If unit sale prices of x, y and z are 2.50, 1.50 and 1.00, respectively, find the total revenue in each market with the help of matrix algebra. (b) If the unit costs of the above three commodities are 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.

Find the matrix X so that 1 2 3 7 8 9X 4 5 6 2 4 6 − − −   =       Choose the correct answer in the following questions:

If A = is such that A² = I, then (A) 1 + α² + βγ = 0 (B) 1 - α² + βγ = 0 (C) 1 - α² - βγ = 0 (D) 1 + α² - βγ = 0

If the matrix A is both symmetric and skew symmetric, then (A) A is a diagonal matrix (B) A is a zero matrix (C) A is a square matrix (D) None of these

If A is square matrix such that A2 = A, then (I + A)³ - 7 A is equal to (A) A (B) I - A (C) I (D) 3A