let A=(3 x)
(-2 -3)
if A=A^-1, what is x?
please show work so i can understand
Is (3 x) a 2 x 1 matrix?
What is the meaning of the second line, (-2 -3)? It just sits there.
Is the A matrix in the first line supposed to operate on (3 x) ?
(3 x) and (-2 -3) cannot be multiplied.
One has ti be a column vector.
The second line is probably meant to be the second row of the 2x2 matrix.
I.e. A=
(3 x}
(-2 -3)
If A=A-1, then
A-1A = I
(3 x} (3 x}
(-2 -3)(-2 -3) = I
Performing the matrix multiplication:
(9-2x 0)
(0 -2x+9) = I
so
9-2x=1
or x=4
Substitute x=4 and redo the matrix multiplication to make sure that the result is a 2x2 identity matrix.
To find the inverse of matrix A, we need to solve the equation: A * A^(-1) = I, where I is the identity matrix.
Given matrix A:
A = [ 3 x ]
[ -2 -3 ]
We need to find its inverse A^(-1) such that A * A^(-1) = I.
Let's calculate the product of A and A^(-1) first:
A * A^(-1) = [ 3 x ] * [ a b ] = [ 1 0 ]
[ -2 -3 ] [ c d ] [ 0 1 ]
Using matrix multiplication, we get the following equations:
3a - 2c = 1 ...(1)
3b - 2d = 0 ...(2)
-xa - 3c = 0 ...(3)
-xb - 3d = 1 ...(4)
Let's solve these equations simultaneously to find the values of a, b, c, and d which represents the inverse matrix.
From equation (1), we can determine a in terms of c:
3a = 1 + 2c
a = (1 + 2c)/3 ...(5)
From equation (2), we can determine b in terms of d:
3b = 2d
b = 2d/3 ...(6)
Substituting equations (5) and (6) into equations (3) and (4), we get:
-x((1 + 2c)/3) - 3c = 0 ...(7)
-x(2d/3) - 3d = 1 ...(8)
Let's solve equations (7) and (8) for x:
From equation (7), we can solve for c in terms of x:
-((1 + 2c)/3) = 3cx
-1 - 2c = 9cx
c(-2 - 9x) = -1
c = -1/(-2 - 9x)
c = 1/(2 + 9x) ...(9)
Substituting equation (9) into equation (8), we can solve for d in terms of x:
-x(2d/3) - 3d = 1
-x(2d)/3 - 3d = 1
(-2x/3 - 3)d = 1
d = 1/(-2x/3 - 3)
d = -3/(2x + 9) ...(10)
Now that we have expressions for c and d in terms of x, we can substitute these values into equations (5) and (6) to obtain expressions for a and b:
a = (1 + 2c)/3
a = (1 + 2(1/(2 + 9x)))/3
a = (1 + 2/(2 + 9x))/3
a = (2 + 2/(2 + 9x))/3
a = (2(2 + 9x) + 2)/(3(2 + 9x))
a = (4 + 18x + 2)/(6 + 27x)
a = (18x + 6)/(6 + 27x) ...(11)
b = 2d/3
b = 2(-3)/(3(2x + 9))
b = -6/(6 + 27x) ...(12)
Finally, we set x = a, since A = A^(-1). From equation (11), we have:
a = (18a + 6)/(6 + 27a)
Now, we can solve for a:
6a + 27a^2 = 18a + 6
27a^2 - 12a = 0
3a(9a - 4) = 0
This equation will have two solutions:
a = 0 (discard this solution because it will not yield an inverse matrix)
9a - 4 = 0
9a = 4
a = 4/9
Therefore, x = a = 4/9.