These eight original questions cover the four matrix skills in order. Write your working on paper first, then open the answer.
Questions
Question 1. State the order of the matrix with rows (2, 5, 1) and (0, −3, 4). Then state the entry in row 2, column 3.
Answer
There are 2 rows and 3 columns, so the order is 2 × 3.
Row 2 is (0, −3, 4), and column 3 of that row is 4.
Question 2. A has rows (4, −2) and (1, 6). B has rows (3, 5) and (−1, 2). Find A + B and A − B.
Answer
Both matrices are 2 × 2, so the operations exist.
A + B: (4 + 3, −2 + 5) and (1 + (−1), 6 + 2), so rows (7, 3) and (0, 8).
A − B: (4 − 3, −2 − 5) and (1 − (−1), 6 − 2), so rows (1, −7) and (2, 4).
Question 3. Find 3P − Q, where P has rows (2, 0) and (1, −1), and Q has rows (4, 3) and (−2, 5).
Answer
3P has rows (6, 0) and (3, −3).
Subtract Q entry by entry: (6 − 4, 0 − 3) and (3 − (−2), −3 − 5).
3P − Q has rows (2, −3) and (5, −8).
Question 4. Find AB, where A has rows (1, 3) and (2, −1), and B has rows (4, 0) and (2, 5).
Answer
Row 1: 1(4) + 3(2) = 10, and 1(0) + 3(5) = 15.
Row 2: 2(4) + (−1)(2) = 6, and 2(0) + (−1)(5) = −5.
AB has rows (10, 15) and (6, −5). Check the orders first: 2 × 2 times 2 × 2 gives 2 × 2.
Question 5. Find the inverse of M, which has rows (4, 3) and (5, 4).
Answer
Determinant = 4(4) − 3(5) = 16 − 15 = 1.
Swap 4 and 4, and negate 3 and 5: rows (4, −3) and (−5, 4).
M⁻¹ has rows (4, −3) and (−5, 4). Check: row 1 of MM⁻¹ is 4(4) + 3(−5) = 1 and 4(−3) + 3(4) = 0, so the identity appears.
Question 6. The matrix with rows (6, 3) and (4, 2) has an inverse. True or false? Explain.
Answer
Determinant = 6(2) − 3(4) = 12 − 12 = 0.
False. The determinant is zero, so the inverse does not exist.
Question 7. Use matrices to solve 2x + 3y = 12 and x + 2y = 7.
Answer
Coefficient matrix A has rows (2, 3) and (1, 2). Determinant = 2(2) − 3(1) = 1.
A⁻¹ has rows (2, −3) and (−1, 2).
x = 2(12) − 3(7) = 24 − 21 = 3. y = −1(12) + 2(7) = 2.
x = 3 and y = 2. Check: 2(3) + 3(2) = 12, and 3 + 2(2) = 7.
Question 8. Three mugs and two plates cost RM25. Two mugs and one plate cost RM14. Form a matrix equation and find the price of a mug and a plate.
Answer
Let m be the price of a mug and p the price of a plate. Then 3m + 2p = 25 and 2m + p = 14.
The coefficient matrix has rows (3, 2) and (2, 1). Determinant = 3(1) − 2(2) = −1.
The inverse is 1 ÷ (−1) times the rows (1, −2) and (−2, 3), which gives rows (−1, 2) and (2, −3).
m = −1(25) + 2(14) = 3. p = 2(25) − 3(14) = 8.
A mug costs RM3 and a plate costs RM8. Check: 3(3) + 2(8) = 25, and 2(3) + 8 = 14.
If you got these wrong
Wrong order or entry in question 1: revisit reading matrix order and entries.
Slips in questions 2 to 4 point to adding and multiplying matrices. Questions 5 and 6 need finding the inverse of a 2 × 2 matrix. Questions 7 and 8 use solving simultaneous equations using matrices.
Log each slip in the mistake log and paper-error review, try your own numbers in the two-by-two matrix operations tutor, and build a timed set with the timed original practice session builder.
For a teacher to go through your working, see online one-to-one Mathematics tuition.