Determinants

Answer

Using formula: det(A) = a₁₁a₂₂ - a₁₂a₂₁ Step 1: Identify elements: a₁₁=2, a₁₂=4, a₂₁=-1, a₂₂=2 Step 2: Apply formula: det = (2)(2) - (4)(-1) Step 3: Calculate: det = 4 + 4 = 8 Answer: 8…

Answer

Step 1: Apply 2×2 formula: det = (x)(x) - (x+1)(x-1) Step 2: Expand first term: x² Step 3: Expand second term using difference of squares: (x+1)(x-1) = x² - 1 Step 4: Substitute: det = x² - (x² - 1) S…

Answer

The determinant a₁₁a₂₂ - a₁₂a₂₁ represents whether the system a₁x + b₁y = c₁ and a₂x + b₂y = c₂ has a unique solution. If det ≠ 0, the lines intersect at exactly one point (unique solution). If det = …

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Expand along a row (or column) to convert a 3×3 determinant into a sum of 2×2 determinants. Choose the row/column with most zeros to minimize calculations. Expansion along row i: det = Σ aᵢⱼ(-1)^(i+j)…

Answer

Step 1: Notice column 3 has two zeros. Expand along C₃. Step 2: det = 4(-1)^(1+3)|−1 3| + 0 + 0 |4 1| Step 3: Calculate 2×2 determinant: (−1)(1) − (3)(4) = −1 − 12 = −13 S…

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Step 1: Expand along R₁: det = 0·M₁₁ − sinα·M₁₂ − cosα·M₁₃ Step 2: Find M₁₂ = |−sinα sinβ| = 0 − sinβ·cosα = −sinβ·cosα |cosα 0 | Step 3: Find M₁₃ = |−sinα 0 | = sinα·sinβ …

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Step 1: Understand minor - delete row 2 and column 3 where element a₂₃=6 lies Step 2: Remaining elements form 2×2 matrix: |1 2| |7 8| Step 3: Calculate minor: M₂₃ = 1(8) − 2(7) = 8 − 14 = −6 Answer:…

Answer

Step 1: Recall formula: Aᵢⱼ = (−1)^(i+j)·Mᵢⱼ Step 2: From previous card, M₂₃ = −6 Step 3: Apply sign factor: A₂₃ = (−1)^(2+3)·(−6) = (−1)^5·(−6) Step 4: Evaluate: (−1)^5 = −1, so A₂₃ = (−1)(−6) = 6 An…