| 2 3 5|
A=| 1 7 9|
|-3 2 10|
,
|3|
B=|4|
|1|
I need to solve this matrix equation
AX = B FOR X BY FINDING A^-1, given A and B as follows. I am new at this did I write A^1 right? This is suppose to be an A with a minus one at the top.
1kw
Given A and B, where A is a 3x3 matrix, and B is a vector of 3, and
AX=B
To solve for X, we would pre-multiply by the inverse of A to give
A-1A X = A-1B
Since A-1A is the identity matrix by definition of the inverse, we end up with
X = A-1B
To find X, it will be necessary to find the inverse of A, premultiply with B and the result (3x1 vector) will be X as required.
The inverse of A can be found by Gauss elimination, or any other method that you have learned.
The value of X is a multiple or sub-multiple of [17,4,7].
Yes, you wrote A^-1 correctly. It represents the inverse of matrix A. To solve the matrix equation AX = B and find matrix X, you can follow these steps:
Step 1: Calculate the inverse of matrix A (A^-1).
To find the inverse of matrix A, you can use the formula A^-1 = (1/det(A)) * adj(A), where det(A) represents the determinant of matrix A, and adj(A) stands for the adjugate of A.
First, let's calculate the determinant of matrix A. The determinant of a 3x3 matrix can be calculated using the following formula:
det(A) = (a11 * a22 * a33) + (a12 * a23 * a31) + (a13 * a21 * a32) - (a31 * a22 * a13) - (a32 * a23 * a11) - (a33 * a21 * a12)
Substituting values from matrix A:
det(A) = (2 * 7 * 10) + (3 * 9 * (-3)) + (5 * 1 * 2) - ((-3) * 7 * 2) - (2 * 9 * 5) - (10 * 1 * (-3))
det(A) = 140 + (-81) + 10 - (-42) - 90 - (-30)
det(A) = 140 - 81 + 10 + 42 - 90 + 30
det(A) = 51
Since det(A) is not zero, matrix A is invertible.
Next, we need to find the adjugate of matrix A. The adjugate of a matrix is the transpose of its cofactor matrix.
The cofactor matrix of matrix A is obtained by calculating the determinants of the minor matrices formed by removing each element of A.
The cofactor matrix of A is:
| (7*10 - 9*2) (-2*10 + 9*3) (2*2 - 7*3)|
| (-(1*10 - 9*(-3))) (2*10 - 9*(-3)) (-2*2 - 7*(-3))|
| (1*9 - 7*(-3)) (-(2*9 - 7*3)) (2*7 - 1*3)|
Simplifying:
| 52 -24 -11|
|-87 36 -17|
| 24 -15 11|
Now, we take the transpose of the cofactor matrix to find the adjugate of A:
| 52 -87 24|
|-24 36 -15|
|-11 -17 11|
Finally, we can calculate the inverse of matrix A by dividing the adjugate of A by the determinant of A:
A^-1 = (1/det(A)) * adj(A)
A^-1 = (1/51) *
| 52 -87 24|
|-24 36 -15|
|-11 -17 11|
Step 2: Calculate matrix X.
To find matrix X, we can multiply both sides of the equation AX = B by the inverse of A.
AX = B
A^-1 * AX = A^-1 * B
I * X = A^-1 * B
X = A^-1 * B
Now substitute the values we calculated:
X =
| 52 -87 24| |3|
|-24 36 -15| * |4|
|-11 -17 11| |1|
To compute X, perform matrix multiplication:
X =
(52*3 + (-87)*4 + 24*1) (-24*3 + 36*4 + (-15)*1) (-11*3 + (-17)*4 + 11*1)
Can you solve the equation to obtain the values for X based on the above calculation?