Matrices

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A matrix is a rectangular arrangement of mn numbers in m rows and n columns, enclosed in [ ] or ( ). Definition: A matrix of order m × n has m rows and n columns. Notation: A = [aij]m×n where aij is

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This is a Scalar Matrix. Step-by-step identification: 1. Check if it's square: Yes (3×3) 2. Check diagonal elements: 5, 5, 5 (all same) 3. Check non-diagonal elements: All are 0 Definition: A diagon

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Step 1: Find A^T by interchanging rows and columns A = [1 2 3; 4 5 6]2×3 A^T = [1 4; 2 5; 3 6]3×2 Step 2: Find (A^T)^T (A^T)^T = [1 2 3; 4 5 6]2×3 Step 3: Verify (A^T)^T = A (A^T)^T = [1 2 3; 4 5 6]

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Step 1: Add the matrices on left side [2x+y+1 1-y+6; 3+3 4y+0] = [3 5; 6 18] [2x+y+1 7-y; 6 4y] = [3 5; 6 18] Step 2: Use equality of matrices Corresponding elements are equal: 2x+y+1 = 3 ... (1) 7-y

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Given: A = [1 2; 3 4]2×2 and B = [5 6; 7 8]2×2 Step 1: Check conformability Columns of A = 2, Rows of B = 2 ✓ (Multiplication possible) Order of AB = 2×2 Step 2: Calculate each element of AB AB = [c

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To prove the matrix is singular, we need to show |A| = 0. A = [x+y y+z z+x; 1 1 1; z x y] Step 1: Calculate determinant |A| = (x+y)[y-x] - (y+z)[y-z] + (z+x)[x-z] Step 2: Expand each term = (x+y)(y

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Definition: A square matrix A = [aij]n×n is symmetric if aij = aji for all i and j. In other words: A = A^T Example: A = [2 3 1; 3 5 4; 1 4 7] Step 1: Check if A^T = A A^T = [2 3 1; 3 5 4; 1 4 7] S

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Step 1: Check if inverse exists |A| = 2(3) - 5(1) = 6 - 5 = 1 ≠ 0 ✓ Step 2: Set up AA^(-1) = I [2 5; 1 3]A^(-1) = [1 0; 0 1] Step 3: Apply row transformations R1 ↔ R2: [1 3; 2 5]A^(-1) = [0 1; 1 0]