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.

Mathematics
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.

ScanSolve AI Answer

Answer

X1=2,X2=3,X3=−4X_1 = 2, X_2 = 3, X_3 = -4

Q1 a) Solve the following set of linear equations by the matrix method. The given system of equations is: X1+2X2+X3=4X_1 + 2X_2 + X_3 = 4 3X1−4X2−2X3=23X_1 - 4X_2 - 2X_3 = 2 5X1+3X2+5X3=−15X_1 + 3X_2 + 5X_3 = -1

Step 1: Write the system in matrix form AX=BAX = B. A=[1213−4−2535],X=[X1X2X3],B=[42−1]A = \begin{bmatrix} 1 & 2 & 1 \\ 3 & -4 & -2 \\ 5 & 3 & 5 \end{bmatrix}, \quad X = \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 \\ 2 \\ -1 \end{bmatrix}

Step 2: Calculate the determinant of matrix AA. det⁡(A)=1((−4)(5)−(−2)(3))−2((3)(5)−(−2)(5))+1((3)(3)−(−4)(5))\det(A) = 1((-4)(5) - (-2)(3)) - 2((3)(5) - (-2)(5)) + 1((3)(3) - (-4)(5)) det⁡(A)=1(−20+6)−2(15+10)+1(9+20)\det(A) = 1(-20 + 6) - 2(15 + 10) + 1(9 + 20) det⁡(A)=1(−14)−2(25)+1(29)\det(A) = 1(-14) - 2(25) + 1(29) det⁡(A)=−14−50+29=−35\det(A) = -14 - 50 + 29 = -35

Step 3: Calculate the adjoint of matrix AA. First, find the cofactor matrix CC:

(-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)) \end{bmatrix}$$ $$C = \begin{bmatrix} -14 & -25 & 29 \\ -7 & 0 & 7 \\ 0 & 5 & -10 \end{bmatrix}$$ The adjoint matrix is the transpose of the cofactor matrix: $$\text{adj}(A) = C^T = \begin{bmatrix} -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 \end{bmatrix}$$ Step 4: Calculate the inverse of matrix $A$. $$A^{-1} = \frac{1}{\det(A)} adj(A) = \frac{1}{-35} \begin{bmatrix} -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 \end{bmatrix}$$ Step 5: Solve for $X$ using $X = A^{-1}B$. $$X = \frac{1}{-35} \begin{bmatrix} -14 & -7 & 0 \\ -25 & 0 & 5 \\ 29 & 7 & -10 \end{bmatrix} \begin{bmatrix} 4 \\ 2 \\ -1 \end{bmatrix}$$ $$X = \frac{1}{-35} \begin{bmatrix} (-14)(4) + (-7)(2) + (0)(-1) \\ (-25)(4) + (0)(2) + (5)(-1) \\ (29)(4) + (7)(2) + (-10)(-1) \end{bmatrix}$$ $$X = \frac{1}{-35} \begin{bmatrix} -56 - 14 + 0 \\ -100 + 0 - 5 \\ 116 + 14 + 10 \end{bmatrix}$$ $$X = \frac{1}{-35} \begin{bmatrix} -70 \\ -105 \\ 140 \end{bmatrix} = \begin{bmatrix} \frac{-70}{-35} \\ \frac{-105}{-35} \\ \frac{140}{-35} \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \\ -4 \end{bmatrix}$$ The solution is: $$\boxed{X_1 = 2, X_2 = 3, X_3 = -4}$$ **Q1 b) Find the values of $x$ for which the matrix $\begin{bmatrix} 2x & -2 \\ -4 & x \end{bmatrix}$ is singular.** Step 1: Understand the condition for a singular matrix. A matrix is singular if its determinant is equal to zero. Step 2: Calculate the determinant of the given matrix. For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$. $$\det \begin{bmatrix} 2x & -2 \\ -4 & x \end{bmatrix} = (2x)(x) - (-2)(-4)$$ $$\det = 2x^2 - 8$$ Step 3: Set the determinant to zero and solve for $x$. $$2x^2 - 8 = 0$$ $$2x^2 = 8$$ $$x^2 = \frac{8}{2}$$ $$x^2 = 4$$ $$x = \pm \sqrt{4}$$ $$\boxed{x = \pm 2}$$ **Q2 a) (i) Evaluate the following $\int_{1}^{2} 4 \cos t \,dt$.** Step 1: Find the antiderivative of $4 \cos t$. The antiderivative of $\cos t$ is $\sin t$. So, the antiderivative of $4 \cos t$ is $4 \sin t$. Step 2: Evaluate the definite integral using the Fundamental Theorem of Calculus. $$\int_{1}^{2} 4 \cos t \,dt = [4 \sin t]_{1}^{2}$$ $$= 4 \sin(2) - 4 \sin(1)$$ $$= 4(\sin(2) - \sin(1))$$ $$\boxed{4(\sin(2) - \sin(1))}$$ (Note: The angles are in radians.) Send me the next one 📸
Was this helpful?

Need help with your own homework?

Get instant step-by-step solutions to any question. Free to start.

Ask Your Question
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp

More Mathematics Questions

QA triangle is formed by the coordinates A (2, 1), B (4, 1) and C (1, 6). Draw the image of the trian…QThe figure below shows a net of a prism whose cross - section is an equilateral triangle. Sketch the…QSimplify p²-2p9+9² P³-p9²+p²9-9³QTwo consecutive even numbers are 5k-5 and 6k-10. Find the numbers.QThe mean age of a second year class of a school is 18 2/5. At the end of the promotion examination,…
See all Mathematics questions →

Still have questions?

Quick Answer

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.

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.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
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 📸