What is the solution of the matrix equation?
[9 4]is the top part of the first matrix, [2 1] is the bottom part of the first matrix. X= is between both matrixes. [-9 -6] is the top part of the second matrix [-1 -8] is the bottom part of the second matrix.
To find the solution of the matrix equation, we need to find the value of X that satisfies the equation:
[9 4] X [-9 -6]
[2 1] = [-1 -8]
The equation can be expanded as follows:
[9x + 2y 4x + y] = [-9 -6]
[-1 -8]
Comparing corresponding elements on both sides of the equation, we get the following system of equations:
9x + 2y = -9 (equation 1)
4x + y = -6 (equation 2)
2x + y = -1 (equation 3)
x + 8y = -8 (equation 4)
We can solve this system of equations using various methods such as substitution or elimination. Let's solve it using the elimination method:
From equations 2 and 3, we can eliminate y by subtracting equation 2 from equation 3:
(2x + y) - (4x + y) = -1 - (-6)
-2x = 5
x = -5/2
Substituting x = -5/2 into equation 3:
2(-5/2) + y = -1
-5 + y = -1
y = 4
So, the solution to the matrix equation is:
X = [-5/2 4]
To find the solution of the matrix equation, we need to solve for the unknown matrix X. The equation can be written as:
⎡9 4⎤ ⎡-9 -6⎤
⎢ ⎥ × X = ⎢ ⎥
⎣2 1⎦ ⎣-1 -8⎦
To solve this, we can use matrix inversion. The general formula for solving a matrix equation AX = B for X, where A is a square matrix, is:
X = A^(-1) × B
Here, A represents the first matrix, X is the unknown matrix, and B is the second matrix. To find the inverse of A (A^(-1)), we need to calculate the determinant of A.
The determinant of a 2x2 matrix [a b; c d] is given by ad - bc.
In this case, the determinant of A is: (9 * 1) - (4 * 2) = 9 - 8 = 1.
Since the determinant is non-zero (1 ≠ 0), we can proceed to find the inverse of A.
The inverse of a 2x2 matrix [a b; c d] is given by:
[ d -b ]
[-c a ]
Using this formula, we can find the inverse of A:
A^(-1) = [ 1 -4 ]
[ -2 9 ]
Now, we can calculate the solution for X by multiplying A^(-1) and B:
X = [ 1 -4 ] × [ -9 -6 ]
[ -2 9 ] [ -1 -8 ]
To perform the matrix multiplication, we multiply corresponding elements in each row of the first matrix with each column of the second matrix, and sum the products.
X = [ (1 * -9) + (-4 * -1) (1 * -6) + (-4 * -8) ]
[ (-2 * -9) + (9 * -1) (-2 * -6) + (9 * -8) ]
Simplifying this further, we get:
X = [ -9 + 4 -6 + 32 ]
[ 18 - 9 12 - 72 ]
X = [ -5 26 ]
[ 9 -60 ]
Therefore, the solution to the matrix equation is:
⎡-5 26⎤
⎣ 9 -60⎦
To find the solution of the matrix equation, we can calculate it by multiplying the inverse of the first matrix with the second matrix.
Step 1: Calculate the inverse of the first matrix.
Let's call the first matrix A.
A = [9 4]
[2 1]
To find the inverse, we can use the formula:
A^(-1) = (1/det(A)) * Adj(A)
where det(A) is the determinant of A, and Adj(A) is the adjugate of A.
Calculate the determinant of A:
det(A) = (9*1) - (4*2) = 9 - 8 = 1
Calculate the adjugate of A:
Adj(A) = [1 -4]
[-2 9]
Calculate the inverse of A:
A^(-1) = (1/1) * [1 -4] = [1 -4]
[-2 9]) [-2 9])
Therefore, the inverse of the first matrix A is:
A^(-1) = [1 -4]
[-2 9]
Step 2: Multiply the inverse of A with the second matrix.
Let's call the second matrix B.
B = [-9 -6]
[-1 -8]
Now, we can multiply A^(-1) with B:
A^(-1) * B = [1 -4] * [-9 -6] = [1(-9) + (-4)(-1) 1(-6) + (-4)(-8)] = [-9 + 4 -6 + 32]
[-2 9] [-1 -8] [-2(-9) + 9(-1) -2(-6) + 9(-8)] [18 - 9 12 - 72]
= [-5 -10]
[9 -60]
Therefore, the solution of the matrix equation X is:
X = [-5 -10]
[9 -60]