Matrices: algebra, multiplication, determinants, inverses, Ax=b practice questions.

Let A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]] (rows shown). Determine the product AB.

• [[2, 1], [4, 3]]
• [[1, 2], [3, 4]]
• [[2, 1], [3, 4]]
• [[0, 2], [3, 0]]

Answer: [[2, 1], [4, 3]]

AB row 1: [1·0+2·1, 1·1+2·0] = [2, 1]; row 2: [3·0+4·1, 3·1+4·0] = [4, 3], giving [[2,1],[4,3]].

Determine the inverse of the matrix A = [[4, 3], [3, 2]] (rows shown).

• [[−2, 3], [3, −4]]
• [[2, −3], [−3, 4]]
• [[−2, −3], [−3, −4]]
• [[4, −3], [−3, 2]]

Answer: [[−2, 3], [3, −4]]

det(A) = 4·2 − 3·3 = −1, so A⁻¹ = (1/−1)[[2, −3], [−3, 4]] = [[−2, 3], [3, −4]].

Given A = [[2, 0], [1, 3]] and B = [[1, 5], [4, 2]] (rows shown), determine A + 2B.

• [[4, 10], [9, 7]]
• [[3, 5], [5, 5]]
• [[4, 10], [6, 8]]
• [[2, 10], [8, 12]]

Answer: [[4, 10], [9, 7]]

2B = [[2, 10], [8, 4]]; adding A gives [[2+2, 0+10], [1+8, 3+4]] = [[4, 10], [9, 7]].

The system Ax = b has A = [[1, 2], [2, 4]] (rows shown). What can be concluded about its solutions?

• det A = 0, so no unique solution
• There is exactly one solution for any b
• A⁻¹ = [[4, −2], [−2, 1]]
• The solution is always x = 0

Answer: det A = 0, so no unique solution

det(A) = 1·4 − 2·2 = 0, so A is singular and A⁻¹ does not exist; the system has either no solution or infinitely many.

Determine the product of the row matrix [2 3] and the column matrix [[4], [5]].

• [23]
• [8, 15]
• [[8], [15]]
• [14]

Answer: [23]

[2 3]·[[4],[5]] = (2)(4) + (3)(5) = 8 + 15 = 23, a 1×1 matrix [23].

Given A = [[1, 2], [0, 1]] (rows shown), determine A².

• [[1, 4], [0, 1]]
• [[1, 2], [0, 1]]
• [[1, 4], [0, 2]]
• [[2, 4], [0, 2]]

Answer: [[1, 4], [0, 1]]

A² row 1: [1·1+2·0, 1·2+2·1] = [1, 4]; row 2: [0·1+1·0, 0·2+1·1] = [0, 1], giving [[1,4],[0,1]].