Linear Algebra Answers

Given

"A=\\left(\\begin{array}{rr}-2&2\\\\2&-4\\end{array}\\right)\\quad \\textrm{and} \\quad B=\\left(\\begin{array}{rr}2&3\\\\2&-1\\end{array}\\right)."

Select the option(s) below which represent "\\left(AB^{-1}\\right)^{-1}."

1. "\\displaystyle \\left(\\begin{array}{rr}-3.5&-2.5\\\\-1.5&-0.5\\end{array}\\right)."
2. "\\displaystyle \\left(\\begin{array}{rr}3.5&2.5\\\\1.5&0.5\\end{array}\\right)."
3. A^-1B
4. B^-1A^-1
5. BA^-1

You are given that A and B are two square matrices of the same order, such that

"\\mathrm{det}\\left(A^{-1}B\\right)=20\\quad \\mathrm{and} \\quad \\mathrm{det} \\left(B\\right)=5."

Which of the following is true?

1. "\\mathrm{det}(A)=-15\\quad \\text{and} \\quad \\mathrm{adj}(A)=-\\frac{1}{15}"
2. "\\mathrm{det}(A)=4\\quad \\text{and} \\quad\\mathrm{adj}(A)=4A^{-1}"
3. "\\mathrm{det}(A)=\\frac{1}{4}\\quad \\text{and} \\quad \\mathrm{adj}(A)=\\frac{1}{4}A^{-1}"
4. "\\mathrm{det}(A)=25\\quad \\text{and} \\quad \\mathrm{adj}(A)=25"
5. "\\mathrm{det}(A)=15\\quad \\text{and} \\quad \\mathrm{adj}(A)=15 A^{-1}"

.

Consider the following System of equations:

3𝑥 + 2𝑦 + 𝑧 = 3

2𝑥 + 𝑦 + 𝑧 = 0

6𝑥 + 2𝑦 + 4𝑧 = 6

a. Apply Gaussian elimination method to reduce the system to triangular form.

b. What do you observe from your answer in part (a) above?

Given that matrix

2 1 3 

A= -1 -2 5

-4 4 -5

, determine element a₃₂.

7.) Find X so that for any 3 × 3 real matrix A you get AX = XA = A [Hint : what property is being exhibited by real number p so that for any real w we get wp = pw = w then interpret for matrices.] 1 8.) Consider K =  1 −1 1 −1  then we get K2 = 0 Does this hold for real numbers? Motivate.

A system of equations is given below, 𝑡𝑥 + 2𝑦 + 3𝑧 = 𝑎 2𝑥 + 3𝑦 − 𝑡𝑧 = 𝑏 3𝑥 + 5𝑦 + (𝑡 + 1)𝑧 = 𝑐 Where 𝑡 is an integer and 𝑎, 𝑏, 𝑐 are real constants. The system does not have a unique solution, but it is consistent. Show that 𝑎 + 𝑏 = 𝑐

Select 3 different digits from these numbers (987621). Use only these numbers as coefficient and create a matrix of 5x5 (with all coefficient non zero) that has rank 3. Also, explain why the rank is equal to 3.

Let A be a 2 x 2 matrix. Show that some non-trivial linear combination of A^4, A^3, A^2, A. and I2 is equal 0. Generalize to n x n matrices. Note that I2 is 2 x 2 identity matrix.