What is the solution of the matrix equation?
[9 4];[2 1] X= [-9 -6];[-1 -8]
To find the solution of the matrix equation, we need to find the inverse of the coefficient matrix and multiply it by the right-hand side matrix.
The coefficient matrix is given by:
[9 4]
[2 1]
To calculate the inverse, we can use the formula for a 2x2 matrix:
[1/det(A)] * [d -b]
[-c a]
where det(A) is the determinant of matrix A, a is the upper left element, b is the upper right element, c is the lower left element, and d is the lower right element.
In this case, a = 9, b = 4, c = 2, and d = 1. Therefore, we have:
det(A) = (9*1) - (4*2) = 9 - 8 = 1
Now, we can calculate the inverse:
[1/det(A)] * [d -b]
[-c a] = [1/1] * [1 -4]
[-2 9] = [1 -4]
[-2 9]
The inverse matrix is:
[1 -4]
[-2 9]
Now, we can multiply the inverse matrix by the right-hand side matrix:
[1 -4] [-9 -6] [(-9*1) + (-6*-2) (-9*-4) + (-6*9)]
[-2 9] * [-1 -8] = [(-2*1) + (9*-2) (-2*-4) + (9*9)]
Simplifying the calculations, we have:
[1 -4] [-9 -6] [-9 + 12 36 - 54]
[-2 9] * [-1 -8] = [-2 -18 -8 + 81]
[1 -4] [-9 -6] [3 -18]
[-2 9] * [-1 -8] = [-20 73]
Therefore, the solution to the matrix equation is:
X = [3 -18]
[-20 73]
To find the solution to the matrix equation [9 4; 2 1] X = [-9 -6; -1 -8], we can use the inverse of the coefficient matrix.
1. Begin by writing the given matrix equation in the form AX = B, where A is the coefficient matrix and X is the unknown matrix.
[9 4; 2 1] X = [-9 -6; -1 -8]
2. Next, we can find the inverse of the coefficient matrix A. Let's denote A^-1 as the inverse of A.
A = [9 4; 2 1]
Using the inverse matrix formula, A^-1 = (1/det(A)) * adj(A), where det(A) represents the determinant of A, and adj(A) represents the adjugate of A.
First, calculate the determinant of A:
det(A) = (9 * 1) - (4 * 2) = 9 - 8 = 1
Next, find the adjugate of A by swapping the main diagonal elements and changing the sign of the off-diagonal elements:
adj(A) = [1 -4; -2 9]
Finally, compute the inverse of A using the determined determinant and adjugate:
A^-1 = (1/1) * [1 -4; -2 9] = [1 -4; -2 9]
3. Now, multiply both sides of the matrix equation by the inverse of A (A^-1) to isolate the unknown matrix X:
A^-1 * A * X = A^-1 * B
Simplifying,
I * X = A^-1 * B
where I is the identity matrix.
X = A^-1 * B
4. Substitute the calculated values of A^-1 and B into the equation to find the solution:
X = [1 -4; -2 9] * [-9 -6; -1 -8]
Performing the matrix multiplication,
X = [1*(-9) + (-4)*(-1) 1*(-6) + (-4)*(-8); -2*(-9) + 9*(-1) -2*(-6) + 9*(-8)]
Simplifying,
X = [-5 -38; -27 -72]
Therefore, the solution to the given matrix equation is:
X = [-5 -38; -27 -72]
To solve the matrix equation [9 4 ; 2 1] X = [-9 -6 ; -1 -8], we can use the inverse of the coefficient matrix.
Step 1: Calculate the inverse of the coefficient matrix [9 4 ; 2 1].
Let A be the coefficient matrix: A = [9 4 ; 2 1]
The inverse of A, denoted as A^(-1), can be found using the formula:
A^(-1) = (1 / det(A)) * Adj(A)
Here, Adj(A) represents the adjugate matrix of A, and det(A) represents the determinant of A.
The adjugate matrix is obtained by taking the transpose of the matrix of cofactors.
To calculate the determinant of A:
det(A) = (9 * 1) - (4 * 2) = 9 - 8 = 1
Next, let's calculate the matrix of cofactors:
The matrix of cofactors is obtained by swapping the sign of each element in the matrix of minors. The matrix of minors is obtained by taking the determinant of each 2x2 submatrix.
Matrix of minors:
[1 -2]
[-2 9]
Matrix of cofactors:
[1 -2]
[-2 9]
Taking the transpose of the matrix of cofactors gives us the adjugate matrix:
Adj(A) = [1 -2 ; -2 9]
Now, we can calculate A^(-1) using the formula mentioned above:
A^(-1) = (1 / 1) * [1 -2 ; -2 9] = [1 -2 ; -2 9]
Step 2: Multiply the inverse of the coefficient matrix with the right-hand side matrix.
Let B be the right-hand side matrix: B = [-9 -6 ; -1 -8]
We can calculate the solution matrix X by multiplying the inverse of A with B:
X = A^(-1) * B
Multiplying the matrices:
X = [1 -2 ; -2 9] * [-9 -6 ; -1 -8]
X = [(-9 * 1) + (-6 * -2) (-9 * -2) + (-6 * 9) ; (-1 * 1) + (-8 * -2) (-1 * -2) + (-8 * 9)]
X = [9 + 12 -18 - 54 ; -1 + 16 2 + 72]
X = [21 -72 ; 15 74]
Therefore, the solution to the matrix equation [9 4 ; 2 1] X = [-9 -6 ; -1 -8] is:
X = [21 -72 ; 15 74]