Matrices quiz

A matrix with 2 rows and 3 columns has order:

The number of all possible 2 x 2 matrices with each entry 0 or 1 is:

If A is 3 x 4 and B is 4 x 2, then AB is of order:

A square matrix A is symmetric if:

The diagonal elements of a skew symmetric matrix are:

For any matrix A, (A^T)^T equals:

If A = [[0, 1], [1, 0]], then A^2 is:

Matrix multiplication is in general:

If A and B are symmetric matrices of the same order, AB - BA is:

The number of elements in a 3 x 3 matrix is:

If A = [[2, 3], [1, 4]], find A^T and verify that (A^T)^T = A. (2 marks)

Give an example of two non-zero 2 x 2 matrices A and B such that AB = O. (2 marks)

Construct a 2 x 2 matrix with elements a_ij = i + j. (2 marks)

If A is a square matrix, show that A + A^T is symmetric. (2 marks)

Find x if [[2x, 3], [1, 4]] = [[8, 3], [1, 4]]. (2 marks)

Show that the matrix [[0, 2], [-2, 0]] is skew symmetric. (2 marks)

If A = [[1, 0], [0, 1]], find A^5. (2 marks)

If A = [[1, 2]] and B = [[3], [4]], find AB and BA and state their orders. (2 marks)

Show that B = [[1, 1], [1, 2]] is the inverse of A = [[2, -1], [-1, 1]]. (2 marks)

If A is a 2 x 2 matrix with A = kI, what type of matrix is A? (2 marks)

Find AB if A = [[1, -2], [2, 3]] and B = [[2, 1], [3, 4]]. (3 marks)

Find X if 2A + 3X = 5B, where A = [[8, 0], [4, -2], [3, 6]] and B = [[2, -2], [4, 2], [-5, 1]]. (3 marks)

Find x if [1 x 1] [[1, 3, 2], [2, 5, 1], [15, 3, 2]] [[1], [2], [x]] = O. (3 marks)

Express A = [[2, 3], [4, 5]] as the sum of a symmetric and a skew symmetric matrix. (3 marks)

If f(x) = x^2 - 5x + 6 and A = [[2, 1], [1, 2]], find f

Read the passage and answer the questions. A school buys chairs, tables and boards for two branches. Branch 1 needs 20 chairs, 10 tables and 2 boards; branch 2 needs 30 chairs, 12 tables and 3 boards. Chairs cost Rs 500, tables Rs 1,200 and boards Rs 3,000. (i) Write the requirement matrix and price matrix. (ii) Find the cost for each branch using matrix multiplication. (iii) Find the total cost. (5 marks)

Read the passage and answer the questions. A coding system uses A = [[2, -1], [-1, 1]] to encode messages and B = [[1, 1], [1, 2]] to decode them. (i) Find AB. (ii) What is the relation between A and B? (iii) Why must the decoding matrix be the inverse of the encoding matrix? (5 marks)

Read the passage and answer the questions. A teacher gives the class the matrix M = [[0, 3, -4], [-3, 0, 5], [4, -5, 0]]. (i) Find M^T. (ii) Is M symmetric or skew symmetric? (iii) What can you say about the diagonal entries of such matrices? (5 marks)

Read the passage and answer the questions. Family F1 has 2 men, 3 women and 1 child; family F2 has 2 men, 2 women and 4 children. Daily needs are 2,400 calories and 45 g protein for a man, 1,900 calories and 55 g protein for a woman, and 1,800 calories and 33 g protein for a child. (i) Write the family matrix and the requirement matrix. (ii) Find the calories needed by each family. (iii) Find the protein needed by each family. (5 marks)

Read the passage and answer the questions. A student claims that for any square matrices A and B, (A + B)^2 = A^2 + 2AB + B^2. (i) Expand (A + B)^2 correctly. (ii) When is the student's formula true? (iii) Give an example where it fails. (5 marks)

If A = [[1, 2, 3], [3, -2, 1], [4, 2, 1]], show that A^3 - 23A - 40I = O. (6 marks)

For A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], verify that (A + B)^2 is not equal to A^2 + 2AB + B^2, and explain why. (6 marks)

Express A = [[3, 3, -1], [-2, -2, 1], [-4, -5, 2]] as the sum of a symmetric and a skew symmetric matrix. (6 marks)

A trust fund has Rs 30,000 to invest in two bonds paying 5% and 7% per year. Using matrix multiplication, find how to divide the money to get an annual interest of Rs 1,800. (6 marks)

If A = [[cos theta, sin theta], [-sin theta, cos theta]], prove by mathematical induction that A^n = [[cos n theta, sin n theta], [-sin n theta, cos n theta]] for all natural numbers n. (6 marks)