Determinants and Inverses

Step 1: Use the determinant formula for 2×2 matrix: For |a b|, det = ad - bc |c d|

Step 2: Identify values: a = 2, b = 3, c = 1, d = 4

Step 3: Calculate: det = (2)(4) - (3)(1) = 8 - 3 = 5

Answer: 5

Step 1: Calculate the determinant: det = (1)(4) - (2)(2) = 4 - 4 = 0

Step 2: Apply invertibility criterion: A matrix is invertible if and only if its determinant ≠ 0

Step 3: Conclusion: Since det = 0, the matrix is NOT invertible (singular)

Answer: Not invertible

Step 1: Calculate the determinant: det = (3)(2) - (1)(5) = 6 - 5 = 1

Step 2: Use the inverse formula for 2×2: A⁻¹ = (1/det)[d -b] [-c a]

Step 3: Identify values: a = 3, b = 1, c = 5, d = 2

Step 4: Apply formula: A⁻¹ = (1/1)[2 -1] [-5 3]

A⁻¹ = [2 -1] [-5 3]

Step 5: Verify (AA⁻¹ = I): [3 1][2 -1] = [6-5 -3+3 ] = [1 0] [5 2][-5 3] [10-10 -5+6] [0 1] ✓

Answer: [2 -1] [-5 3]

Step 1: Write in matrix form AX = B: [3 1][x] = [7 ] [5 2][y] [12]

Step 2: Find A⁻¹ (from previous problem): A⁻¹ = [2 -1] [-5 3]

Step 3: Solve X = A⁻¹B: [x] = [2 -1][7 ] [y] [-5 3][12]

Step 4: Calculate: x = 2(7) + (-1)(12) = 14 - 12 = 2 y = -5(7) + 3(12) = -35 + 36 = 1

Step 5: Verify in original equations: 3(2) + 1 = 7 ✓ 5(2) + 2(1) = 12 ✓

Answer: x = 2, y = 1

Step 1: Recognize upper triangular matrix: A matrix where all entries below the main diagonal are zero

Step 2: Use triangular matrix property: The determinant of a triangular matrix is the product of diagonal elements

Step 3: Calculate: det = (1)(4)(6) = 24

Step 4: General expansion verification (if needed): Expand along first column: det = 1|4 5| - 0 + 0 |0 6| = 1(24 - 0) = 24 ✓

Answer: 24