Matrix Operations and Solving Systems Worksheet

Question 1

Let A = [[2, 3], [1, 4]] and B = [[5, -1], [0, 2]]. Find A + B.

Accepted Answer

The sum is [[7, 2], [1, 6]] because corresponding entries are added: 2 + 5 = 7, 3 + (-1) = 2, 1 + 0 = 1, and 4 + 2 = 6.

Question 2

Let A = [[7, 2], [-3, 5]] and B = [[4, -1], [6, 3]]. Find A - B.

Accepted Answer

The difference is [[3, 3], [-9, 2]] because corresponding entries are subtracted: 7 - 4 = 3, 2 - (-1) = 3, -3 - 6 = -9, and 5 - 3 = 2.

Question 3

Let A = [[1, -2], [3, 0]] and B = [[4, 5], [-1, 2]]. Find 3A.

Accepted Answer

The scalar product is [[3, -6], [9, 0]] because each entry of A is multiplied by 3.

Question 4

Let A = [[2, 1], [0, -1]] and B = [[3, 4], [5, 2]]. Find AB.

Accepted Answer

The product is [[11, 10], [-5, -2]]. This comes from row-column multiplication: 2(3) + 1(5) = 11, 2(4) + 1(2) = 10, 0(3) + (-1)(5) = -5, and 0(4) + (-1)(2) = -2.

Question 5

Let A = [[2, 1], [0, -1]] and B = [[3, 4], [5, 2]]. Find BA.

Accepted Answer

The product is [[6, -1], [10, 3]]. This shows that matrix multiplication is not commutative because BA is different from AB.

Question 6

Find the determinant of the matrix [[6, 2], [3, 5]].

Accepted Answer

The determinant is 24 because for a 2 by 2 matrix [[a, b], [c, d]], the determinant is ad - bc. Here, 6(5) - 2(3) = 30 - 6 = 24.

Question 7

Find the determinant of the matrix [[4, -2], [6, 1]].

Accepted Answer

The determinant is 16 because 4(1) - (-2)(6) = 4 + 12 = 16.

Question 8

Find the inverse of the matrix [[2, 1], [5, 3]], if it exists.

Accepted Answer

The inverse exists and is [[3, -1], [-5, 2]] because the determinant is 2(3) - 1(5) = 1, and the inverse of [[a, b], [c, d]] is 1 divided by the determinant times [[d, -b], [-c, a]].

Question 9

Determine whether the matrix [[4, 2], [2, 1]] is invertible. Explain your answer.

Accepted Answer

The matrix is not invertible because its determinant is 4(1) - 2(2) = 4 - 4 = 0. A matrix with determinant 0 does not have an inverse.

Question 10

Write the system x + 2y = 5 and 3x - y = 4 in the matrix equation form AX = B.

Accepted Answer

The matrix form is [[1, 2], [3, -1]] [[x], [y]] = [[5], [4]]. The coefficient matrix is A = [[1, 2], [3, -1]], the variable matrix is X = [[x], [y]], and the constant matrix is B = [[5], [4]].

Question 11

Use inverse matrices to solve the system 2x + y = 7 and x - y = 2.

Accepted Answer

The solution is x = 3 and y = 1. In matrix form, A = [[2, 1], [1, -1]] and B = [[7], [2]]. The inverse of A is 1 divided by -3 times [[-1, -1], [-1, 2]], which simplifies to [[1/3, 1/3], [1/3, -2/3]]. Then X = A^(-1)B = [[3], [1]].

Question 12

Use matrices to solve the system 4x + 3y = 18 and 2x - y = 2.

Accepted Answer

The solution is x = 3 and y = 2. Writing the system in matrix form and solving gives the ordered pair (3, 2). You can check by substitution: 4(3) + 3(2) = 18 and 2(3) - 2 = 4.

Question 13

If A = [[1, 2], [0, 3]] and X = [[x], [y]], solve AX = [[5], [6]].

Accepted Answer

The solution is x = 1 and y = 2. The matrix equation represents the system x + 2y = 5 and 3y = 6. From 3y = 6, y = 2, and then x + 4 = 5, so x = 1.

Question 14

Find the product of the row matrix [[2, -1, 3]] and the column matrix [[4], [0], [-2]].

Accepted Answer

The product is -2 because 2(4) + (-1)(0) + 3(-2) = 8 + 0 - 6 = 2.

Question 15

A theater sold 120 tickets for a total of 1020 dollars. Adult tickets cost 10 dollars and student tickets cost 7 dollars. Let a be the number of adult tickets and s be the number of student tickets. Use a system and solve it with matrices.

Accepted Answer

The system is a + s = 120 and 10a + 7s = 1020. Solving gives a = 60 and s = 60. This means 60 adult tickets and 60 student tickets were sold.

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