plss help me with my math problem involving matrix??

Question

Question – 4: Consider the matrices,A^-1 =(
matrix(
3, 4,
-1, -1
)
)
and,B=(
matrix(
1, 6,
2, -3
)
)


(a) Find the matrix A, then compute the matrix, 2A – 3B.

(b) Use the matrix method to solve the system of equations:{3x + 4y = 2 ,-x -4 = 4}.

(c) Is det[(A+B)^3 ]= [det(A+B)]^3? Verify your answer.

Answer 1

(a)

A^(-1) = matrix( 3, 4; -1, -1)

and

B = matrix(1, 6; 2, -3)



You have the inverse of A and (A^(-1))^(-1) = A



For a 2*2 matrix, M = matrix(a,b;c,d)

M^(-1) = 1/(ad - bc)*matrix(d,-b;-c,a)



We have (A^(-1))^(-1) = A = matrix(-1, -4; 1, 3)



2A = matrix(-2, -8; 2, 6)

3B = matrix(3,18; 6,-9)



2A - 3B = matrix(-2-3,-8-18; 2-6,6--9)

=matrix(-5,-26; -4,15)



(b)

The equations can be represents by the matrix A^-1 (the original matrix given in the question) and a column vector.



(A^-1)*matrix(x; y) = matrix(2;4)



If we premultiply by A then:



A(A^-1)*matrix(x; y) = A * matrix(2;4)



matrix(x; y) = A * matrix(2;4)



So,

matrix(x; y) = matrix(-1, -4; 1, 3) * matrix(2;4)

matrix(x; y) = matrix(-1*2 + -4*4 ; 1*2 + 3*4)

matrix(x; y) = matrix(-18 ; 14)



(c)

A + B = matrix(0, 2; 3, 0)



So the determinant is 0*0 - 2*3 = -6

det(A+B)³ = -216



(A + B)³ = matrix(0, 2; 3, 0)*matrix(0, 2; 3, 0)*matrix(0, 2; 3, 0)

(A + B)³ = matrix(6, 0; 0, 6)*matrix(0, 2; 3, 0)

(A + B)³ = matrix(0, 12; 18, 0)



det((A + B)³ ) = 0 - 12*18 = -216



det((A + B)³ ) = (det(A+B))³



I hope this helps.

Answer 2

Hi,



If A^-1 is (3, 4, -1, -1), then A is its inverse:



[.-1..-4.]

[...1...3.]



If this is A, then 2A - 3B =



[.-5..-26.]

[..-4..15.]



rref [ 3..4..2.]

......[.-1.-1..4.] gives an answer of:



[1..0..-18.]

[.0.1....14.] so x = -18 and y = 14.



Matrix A + B =



[..0..2.]

[..3..0.]



That matrix cubed is:



[.0..12.]

[.18..0.]



Its determinant is -216.



On the other hand, matrix A + B =

[..0..2.]

[..3..0.]

so its determinant is -6 and that determinant cubed is also -216.



This was verified by use of the calculator.



I hope that helps!!

Answer 3

we have both or these :

A^-1X A=

1 0

0 1

==>3x+4y=1

-x-y=0



3x +4y=0

-x-y=1

A

-1 -4

1 3

2A-3B=

(

-2-8

2 6)

-(3 18

6 -9)

= -5 -26

-4 15

Answer 4

Matrix multiplication is purely taking the dot made from row i of matrix A with column j of matrix B, to style the ij get right of entry to of the matrix (AB). think of "row dot column". The order of matrix is (rows)x(columns). If there are 3 rows in A and 3 columns in B, then there might desire to be 3 rows and 3 columns in (AB) for the above dot product to paintings out. additionally, the size of each and every row of A (the variety of columns of A) and the top of each and every row of B (the variety of rows of B) might desire to be equivalent for the dot product to be defined. In all, right here is the guideline governing matrix multiplication orders: (nxm)x(mxp) = (nxp)