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VOOZH | about |
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VOOZH | about |
The inverse of a 2 × 2 matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. It is an important concept in linear algebra and is used to find the solution of a system of linear equations.
The inverse of a 2x2 matrix is one that, when multiplied by the original matrix, results in the identity matrix. This is how the inverse of a matrix is defined for all matrices, not just 2x2.
For any two 2×2 matrix A and B, if A · B = I, where I is identity matrix of 2x2 then we say inverse of matrix exist.
For example, and are two matrices.
Thus, B is the inverse of A.
Some of the terms used in the calculation of the inverse of a 2 × 2 matrix are:
To find the inverse of a 2×2 matrix, follow these steps:
Step 1: For a matrix , calculate the determinant det(A) using the formula: det(A) = ad - bc.
Step 2: If the determinant det(A) is non-zero, proceed; if it's zero, the matrix does not have an inverse.
Step 3: Swap the positions of (a) and (d), change the signs of (b) and (c), and divide the resulting matrix by the determinant:
Step 4: This resulting matrix is the inverse of the original 2×2 matrix.
Example: For the 2 × 2 Matrix , find the inverse.
Solution:
The inverse matrix, A-1, is calculated as follows:
So, for the matrix:
The inverse A-1 would be:
After performing the calculations, you would get the specific values for A-1
To find the inverse of a 2x2 matrix using row operations, follow these steps:
Step 1: Start with the given 2x2 matrix and write it augmented with the identity matrix:
Step 2: Perform row operations to transform the left side of the augmented matrix into the identity matrix. This involves using elementary row operations like swapping rows, multiplying rows by constants, and adding multiples of one row to another.
Step 3: Once the left side becomes the identity matrix, the right side will be the inverse of the original matrix.
Step 4: Express the resulting matrix as the inverse matrix , where (e), (f), (g), and (h) are the entries of the inverse matrix.
Step 5: Finally, verify the result by multiplying the original matrix by its inverse. The result should be the identity matrix.
Example: Find the inverse of the matrix (A):.
Solution:
To find (A-1) such that (AA-1 = I), where (I) is the identity matrix.
Row 1 divided by 2
Row 2 - Row 1
Row 2 × (2/5)
Row 1 - (3/2) × Row 2
The left side is now the identity matrix, and the right side is A-1:
So, the inverse of matrix (A)
The formula for finding the inverse (A⁻¹) of a 2 × 2 matrix is given as follows:
Where ad-bc is the determinant of the given matrix.
One of the most common applications of the inverse of a 2x2 matrix is solving systems of equations in two variables.
To solve a system of 2x2 equations using the inverse matrix method, follow these steps:
Step 1: Write the System of Equations: Represent the system of equations in the form Ax=B, where A is the coefficient matrix, x is the column matrix of variables, and B is the column matrix of constants.
Example:
Given Equation are: 2x + 3y = 8 and 4x - y =6
Step 1: Write the system of equations in matrix form:
Ax = B
Where A = is the coefficient matrix, x = is the column matrix of variables, and
B = is the column matrix of constants.
Step 2: Find the inverse of matrix (A).
To find the inverse, we first calculate the determinant of (A):
det(A) = (2 × -1) - (3 × 4) = -2 - 12 = -14
Since the determinant is non-zero (-14 ≠ 0), the inverse of (A) exists.
Now, the inverse of (A) is given by:
where (a), (b), (c), and (d) are the elements of the cofactor matrix of (A).
Step 3: Matrix Multiplication to find x
Step 4: Verify Solutions
Substitute x = 13/7 and y = 5/7 into the original equations:
2 · 13/7 + 3 · 5/7 = 8
4 · 13/7 - 5/7 = 6
Example 1: Determine if the matrix is invertible. If yes, find its inverse.
Solution:
To determine if the matrix is invertible, we need to calculate its determinant:
|A| = (5 × 3) - (2 × 1) = 15 - 2 = 13
Since the determinant |A| = 13 is non-zero, the matrix is invertible.
To find the inverse of the matrix, we use the formula for the inverse of a 2x2 matrix:
where adj(A) is the adjugate (or adjoint) of matrix (A)
The adjugate of a 2x2 matrix is:
So, for our given matrix
Now, the inverse:
=
Therefore, the inverse of the matrix is
Example 2: Given the matrix , express conditions on (a, b, c) and (d) for the matrix to be invertible.
Solution:
For the given matrix to be invertible, the determinant ( |A| ) must be non-zero. The determinant of a 2x2 matrix is calculated as ( ad - bc ).
So, the condition for the matrix to be invertible is that ( ad - bc ≠ 0)
In other words:
- If ( ad - bc ≠ 0), the matrix is invertible.
- If ( ad - bc = 0 ), the matrix is not invertible (singular).
Example 3: Let . Determine the values of (k) for which the matrix (A) is not invertible. If invertible, find the inverse.
Solution:
To find the values of ( k ) for which the matrix ( A ) is not invertible, we need to calculate its determinant and set it equal to zero.
Given matrix
The determinant |A| = (1 × k) - (2 × 3) = k - 6
For the matrix ( A ) to be not invertible, |A| = 0:
k - 6 = 0
k = 6
∴ the matrix ( A ) is not invertible when ( k = 6 ).
Q1. Find the inverse of the matrix
Q2. Determine the inverse of the matrix
Q3. Compute the inverse of the matrix
Q4. Find the inverse of the matrix
Q5. Calculate the inverse of the matrix
