Q1. a) Solve the following set of linear equations by the matrix method. X1+2X2+X3=4, 3X1-4X2-2X3=2, 5X1+3X2+5X3=-1. b) Find the values of x for which the matrix [[2x, -2],[-4, x]] is singular.

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Asked on August 2, 2026Mathematics

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Accepted Answer · 2026-08-02T03:09:53.781Z

Q1 a) Solve the following set of linear equations by the matrix method. The given system of equations is: X_1 + 2X_2 + X_3 = 4 3X_1 - 4X_2 - 2X_3 = 2 5X_1 + 3X_2 + 5X_3 = -1 Step 1: Write the system in matrix form AX = B. A = 1 & 2 & 1 \\ 3 & -4 & -2 \\ 5 & 3 & 5 , X = X_1 \\ X_2 \\ X_3 , B = 4 \\ 2 \\ -1 Step 2: Calculate the determinant of matrix A. (A) = 1((-4)(5) - (-2)(3)) - 2((3)(5) - (-2)(5)) + 1((3)(3) - (-4)(5)) (A) = 1(-20 + 6) - 2(15 + 10) + 1(9 + 20) (A) = 1(-14) - 2(25) + 1(29) (A) = -14 - 50 + 29 = -35 Step 3: Calculate the adjoint of matrix A. First, find the cofactor matrix C: C = (-4)(5)-(-2)(3) & -((3)(5)-(-2)(5)) & (3)(3)-(-4)(5) \\ -((2)(5)-(1)(3)) & (1)(5)-(1)(5) & -((1)(3)-(2)(5)) \\ ((2)(-2)-(1)(-4)) & -((1)(-2)-(1)(3)) & ((1)(-4)-(2)(3)) C = -14 & -25 & 29 \\ -7 & 0 & 7 \\ 0 & 5 & -10 The adjoint matrix is the transpose of the cofactor matrix: adj(A) = C^T = -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 Step 4: Calculate the inverse of matrix A. A^-1 = (1)/((A)) adj(A) = (1)/(-35) -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 Step 5: Solve for X using X = A^-1B. X = (1)/(-35) -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 4 \\ 2 \\ -1 X = (1)/(-35) (-14)(4) + (-7)(2) + (0)(-1) \\ (-25)(4) + (0)(2) + (5)(-1) \\ (29)(4) + (7)(2) + (-10)(-1) X = (1)/(-35) -56 - 14 + 0 \\ -100 + 0 - 5 \\ 116 + 14 + 10 X = (1)/(-35) -70 \\ -105 \\ 140 = (-70)/(-35) \\ (-105)/(-35) \\ (140)/(-35) = 2 \\ 3 \\ -4 The solution is: X_1 = 2, X_2 = 3, X_3 = -4 Q1 b) Find the values of x for which the matrix 2x & -2 \\ -4 & x is singular. Step 1: Understand the condition for a singular matrix. A matrix is singular if its determinant is equal to zero. Step 2: Calculate the determinant of the given matrix. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. 2x & -2 \\ -4 & x = (2x)(x) - (-2)(-4) = 2x^2 - 8 Step 3: Set the determinant to zero and solve for x. 2x^2 - 8 = 0 2x^2 = 8 x^2 = (8)/(2) x^2 = 4 x = ± sqrt(4) x = ± 2 Q2 a) (i) Evaluate the following _1^2 4 t \,dt. Step 1: Find the antiderivative of 4 t. The antiderivative of t is t. So, the antiderivative of 4 t is 4 t. Step 2: Evaluate the definite integral using the Fundamental Theorem of Calculus. _1^2 4 t \,dt = [4 t]_1^2 = 4 (2) - 4 (1) = 4((2) - (1)) 4((2) - (1)) (Note: The angles are in radians.) Send me the next one 📸

Accepted Answer · 2026-08-02T03:09:53.781Z

Q1 a) Solve the following set of linear equations by the matrix method. The given system of equations is: X_1 + 2X_2 + X_3 = 4 3X_1 - 4X_2 - 2X_3 = 2 5X_1 + 3X_2 + 5X_3 = -1 Step 1: Write the system in matrix form AX = B. A = 1 & 2 & 1 \\ 3 & -4 & -2 \\ 5 & 3 & 5 , X = X_1 \\ X_2 \\ X_3 , B = 4 \\ 2 \\ -1 Step 2: Calculate the determinant of matrix A. (A) = 1((-4)(5) - (-2)(3)) - 2((3)(5) - (-2)(5)) + 1((3)(3) - (-4)(5)) (A) = 1(-20 + 6) - 2(15 + 10) + 1(9 + 20) (A) = 1(-14) - 2(25) + 1(29) (A) = -14 - 50 + 29 = -35 Step 3: Calculate the adjoint of matrix A. First, find the cofactor matrix C: C = (-4)(5)-(-2)(3) & -((3)(5)-(-2)(5)) & (3)(3)-(-4)(5) \\ -((2)(5)-(1)(3)) & (1)(5)-(1)(5) & -((1)(3)-(2)(5)) \\ ((2)(-2)-(1)(-4)) & -((1)(-2)-(1)(3)) & ((1)(-4)-(2)(3)) C = -14 & -25 & 29 \\ -7 & 0 & 7 \\ 0 & 5 & -10 The adjoint matrix is the transpose of the cofactor matrix: adj(A) = C^T = -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 Step 4: Calculate the inverse of matrix A. A^-1 = (1)/((A)) adj(A) = (1)/(-35) -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 Step 5: Solve for X using X = A^-1B. X = (1)/(-35) -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 4 \\ 2 \\ -1 X = (1)/(-35) (-14)(4) + (-7)(2) + (0)(-1) \\ (-25)(4) + (0)(2) + (5)(-1) \\ (29)(4) + (7)(2) + (-10)(-1) X = (1)/(-35) -56 - 14 + 0 \\ -100 + 0 - 5 \\ 116 + 14 + 10 X = (1)/(-35) -70 \\ -105 \\ 140 = (-70)/(-35) \\ (-105)/(-35) \\ (140)/(-35) = 2 \\ 3 \\ -4 The solution is: X_1 = 2, X_2 = 3, X_3 = -4 Q1 b) Find the values of x for which the matrix 2x & -2 \\ -4 & x is singular. Step 1: Understand the condition for a singular matrix. A matrix is singular if its determinant is equal to zero. Step 2: Calculate the determinant of the given matrix. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. 2x & -2 \\ -4 & x = (2x)(x) - (-2)(-4) = 2x^2 - 8 Step 3: Set the determinant to zero and solve for x. 2x^2 - 8 = 0 2x^2 = 8 x^2 = (8)/(2) x^2 = 4 x = ± sqrt(4) x = ± 2 Q2 a) (i) Evaluate the following _1^2 4 t \,dt. Step 1: Find the antiderivative of 4 t. The antiderivative of t is t. So, the antiderivative of 4 t is 4 t. Step 2: Evaluate the definite integral using the Fundamental Theorem of Calculus. _1^2 4 t \,dt = [4 t]_1^2 = 4 (2) - 4 (1) = 4((2) - (1)) 4((2) - (1)) (Note: The angles are in radians.) Send me the next one 📸

Q1. a) Solve the following set of linear equations by the matrix method. X1+2X2+X3=4, 3X1-4X2-2X3=2, 5X1+3X2+5X3=-1. b) Find the values of x for which the matrix [[2x, -2],[-4, x]] is singular.

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Q1. a) Solve the following set of linear equations by the matrix method. X1+2X2+X3=4, 3X1-4X2-2X3=2, 5X1+3X2+5X3=-1. b) Find the values of x for which the matrix [[2x, -2],[-4, x]] is singular.

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