Matrix Equations Systems Test Answer Key with Scoring – Precalculus

Question 1 From Equations to Matrices (3 points)

(a) A has rows (3, −5) and (2, 1), X is the column (x, y) and B is the column (−4, 7).

(b) det A = 3(1) − (−5)(2) = 13.

(c) Since 13 ≠ 0, A is invertible and the system has exactly one solution.

Answer: (a) as stated; (b) 13; (c) yes, unique

Scoring: 1 point for each part.

Question 2 Solve Using an Inverse (3 points)

A has rows (3, 2) and (7, 4); det A = 12 − 14 = −2. So A⁻¹ = −12 times rows (4, −2) and (−7, 3).

X = −12(4·4 − 2·6, −7·4 + 3·6) = −12(4, −10) = (−2, 5).

Check: 3(−2) + 2(5) = 4 and 7(−2) + 4(5) = 6.

Answer: x = −2 and y = 5

Scoring: 1 point for A⁻¹, 1 point for the multiplication, 1 point for the check or the correct solution.

Question 3 Type of Solution Set (2 points)

(a) det = 4(9) − (−6)(−6) = 36 − 36 = 0.

(b) The second equation is −1.5 times the first (4(−1.5) = −6, −6(−1.5) = 9, 10(−1.5) = −15), so both equations describe the same line. There are infinitely many solutions.

Answer: det = 0; infinitely many solutions

Scoring: 1 point for each part.

Question 4 Images of Points (2 points)

(4, −2): 2(4) + (−1)(−2) = 10 and 0(4) + 3(−2) = −6, giving (10, −6).

(−1, 5): 2(−1) + (−1)(5) = −7 and 0 + 3(5) = 15, giving (−7, 15).

Answer: (10, −6) and (−7, 15)

Scoring: 1 point for each image.

Question 5 A Triangle Transformed (3 points)

(a) (0, 1) goes to (2(0) + 0, 1(0) + 1(1)) = (0, 1). (4, 1) goes to (8, 4 + 1) = (8, 5). (4, 3) goes to (8, 4 + 3) = (8, 7).

(b) det = 2(1) − 0(1) = 2. The original area is 12(4)(2) = 4, so the image area is 4 · 2 = 8 square units.

Answer: (0, 1), (8, 5), (8, 7); det = 2; area 8

Scoring: 1 point for part (a); 1 point for the determinant and 1 point for the area.

Question 6 Matrices from Descriptions (2 points)

(a) (1(−3) + 0, 0 − 1(7)) = (−3, −7).

(b) A 90° clockwise rotation sends (1, 0) to (0, −1) and (0, 1) to (1, 0). Writing these as columns gives rows (0, 1) and (−1, 0).

Answer: (a) (−3, −7); (b) rows (0, 1) and (−1, 0)

Scoring: 1 point for each part.

Question 7 Snack Bar Prices (3 points)

(a) A has rows (6, 2) and (5, 3), X = (b, j) and B = (18, 17).

(b) det A = 18 − 10 = 8, so A⁻¹ = 18 times rows (3, −2) and (−5, 6). b = 3·18 − 2·178 = 208 = 2.50 and j = −5·18 + 6·178 = 128 = 1.50.

Check: 6(2.50) + 2(1.50) = 18 and 5(2.50) + 3(1.50) = 17.

Answer: bagels $2.50 and juice $1.50

Scoring: 1 point for part (a); 2 points for part (b): 1 for the inverse and 1 for the solution.

Question 8 Singular System (2 points)

det = 3(4) − k(2) = 12 − 2k = 0 gives k = 6.

With k = 6: 3x + 6y = 9 simplifies to x + 2y = 3, and 2x + 4y = 5 simplifies to x + 2y = 2.5. The same left side cannot equal 3 and 2.5, so the lines are parallel and there is no solution.

Answer: k = 6; no solution

Scoring: 1 point for k, 1 point for the classification.