Algebra

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A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Each member of this arrangement is called an element of the matrix. Example: A = [2 3]

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The order of a matrix is expressed as m × n, where 'm' is the number of rows and 'n' is the number of columns. It is read as 'm by n'. Example: Matrix A = [1 2 3] [4 5 6] has

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A square matrix is a matrix in which the number of rows equals the number of columns (m = n). Example: B = [2 -1 3] [0 4 -2] [1 5 7] This is a 3×3 square matrix.

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An identity matrix (I) is a square matrix where: - All diagonal elements equal 1 (aᵢᵢ = 1) - All non-diagonal elements equal 0 (aᵢⱼ = 0 for i ≠ j) - For any matrix A: AI = IA = A (multiplicative ident

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Matrix addition conditions: - Both matrices must have the same order (m×n) - Add corresponding elements: C = A + B where cᵢⱼ = aᵢⱼ + bᵢⱼ Example: [2 3] + [1 4] = [3 7] [1 5] [2 6] [

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Scalar multiplication: For matrix A and scalar k, kA is obtained by multiplying each element of A by k. kA = k[aᵢⱼ] = [kaᵢⱼ] Example: 3[2 1] = [6 3] [4 -2] [12 -6] The order remains u

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Conditions: For matrices A(m×n) and B(p×q), multiplication AB is possible only if n = p. Process: Element cᵢₖ of product matrix C = AB is: cᵢₖ = aᵢ₁b₁ₖ + aᵢ₂b₂ₖ + ... + aᵢₙbₙₖ Resultant matrix order

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Transpose A' of matrix A is obtained by interchanging rows and columns. Properties: - (A')' = A - (A + B)' = A' + B' - (kA)' = kA' - (AB)' = B'A' Example: If A = [1 2], then A' = [1 3]