Consider the linear system x1 3x2 x1 + 4x = 2 3x3 = 1 x2 +...

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Consider the linear system x1 3x2 x1 + 4x₂ = 2 3x3 = 1 x2 + 5x3 = 0 - (a) Write down the coefficient matrix A and compute its determinant. (b) Let a,, i = 1,2,3 denote the row-vectors of A, ordered from top to bottom. Compute the vector cross-products a₁ × a3, and a2 × a3, then the scalar triple-products a₁ (a2 x a3) and a2 (a1 x a3). . (c) Using the results obtained in (b), find vectors u and v that satisfy the matrix-vector equations Aue₁ and Av = e2 where e ,1 1,2,3 denote the standard coordinate vectors in R³. Then find real coefficients A and so that the linear combination X₁ = \u + µv satisfies the above system of equations AX= b where b = (2,1,0). (d) Is this solution unique? Explain your answer in terms of the solution space So for the corresponding homogeneous system AX = 0. Consider the linear system x1 3x2 x1 + 4x₂ = 2 3x3 = 1 x2 + 5x3 = 0 - (a) Write down the coefficient matrix A and compute its determinant. (b) Let a,, i = 1,2,3 denote the row-vectors of A, ordered from top to bottom. Compute the vector cross-products a₁ × a3, and a2 × a3, then the scalar triple-products a₁ (a2 x a3) and a2 (a1 x a3). . (c) Using the results obtained in (b), find vectors u and v that satisfy the matrix-vector equations Aue₁ and Av = e2 where e ,1 1,2,3 denote the standard coordinate vectors in R³. Then find real coefficients A and so that the linear combination X₁ = \u + µv satisfies the above system of equations AX= b where b = (2,1,0). (d) Is this solution unique? Explain your answer in terms of the solution space So for the corresponding homogeneous system AX = 0.

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