Finds inverse reciprocal 2 x 2 matrix using this formula \[ \frac{1}{A} \times \begin{pmatrix} d \ \ \ -b \\ -c \ \ \ a \end{pmatrix} \]
The inverse of a 2x2 matrix is another 2x2 matrix that, when multiplied with the original matrix, produces the identity matrix.
Let's take a 2x2 matrix A:
[A] =
[a b]
[c d]
The inverse of matrix A, denoted as A^(-1), can be found using the following formula:
A^(-1) = (1 / (ad - bc)) * [d -b] [-c a]
Where ad - bc is the determinant of matrix A, if the determinant is non-zero.
If the determinant is zero, then the matrix A is said to be non-invertible or singular, and it does not have an inverse.
To use the MathCrave matrix calculator, you enter the values of the 2x2 matrix into the calculator, and it will calculate and display the inverse matrix. This can be useful in various mathematical and engineering applications, such as solving systems of linear equations or finding the coefficients of transformations.
For a 2 by 2 matrix, A = [a b; c d], the formula for finding its inverse is as follows:
A^-1 = (1 / det(A)) * [d -b; -c a],
where det(A) is the determinant of matrix A.
Calculate the determinant of the given matrix A using the formula: det(A) = ad - bc.
Determine if the matrix is invertible. If det(A) ≠ 0, then A is invertible.
Compute the reciprocal of the determinant: 1 / det(A).
Swap the elements in the main diagonal of the matrix:
i. Swap a and d.
ii. Change the sign of b and c.
Multiply each element in the matrix by the reciprocal of the determinant calculated in step 3.
Find the inverse of matrix A = [3 4; 1 2].
Step 1: Calculate the determinant of matrix A:
det(A) = (3 2) - (4 1)
= 6 - 4
= 2
Step 2: Determine if the matrix is invertible:
Since det(A) = 2 ≠ 0, matrix A is invertible.
Step 3: Compute the reciprocal of the determinant:
1 / det(A) = 1 / 2 = 0.5
Step 4: Swap the elements in the main diagonal and change the signs of b and c:
A^-1 = (0.5) * [2 -4; -1 3]
Step 5: Multiply each element in the matrix by the reciprocal of the determinant:
A^-1 = [0.5 2 0.5 (-4); 0.5 (-1) 0.5 3]
= [1 -2; -0.5 1.5]
Therefore, the inverse of matrix
A = [3 4; 1 2] is A^-1
= [1 -2; -0.5 1.5].
The formula for finding the inverse of a 2 by 2 matrix is derived from Cramer's Rule, an important theorem in linear algebra. Cramer's Rule provides a method to solve a system of linear equations by using determinants. It states that if a system of n linear equations in n unknowns has a unique solution, then the coefficient matrix of the system is invertible.
1. Find the inverse of the matrix A = [[3, 4], [2, 5]].
2. Verify if the matrix B = [[2, -3], [-4, 6]] is the inverse of A = [[3, 4], [2, 5]].
3. Find the inverse of the matrix C = [[1, 2], [3, 4]].
4. Verify if the matrix D = [[0.5, -1], [-1.5, 2]] is the inverse of C = [[1, 2], [3, 4]].
5. Find the inverse of the matrix E = [[-1, 2], [4, -3]].
6. Verify if the matrix F = [[1.5, -1], [2, -2]] is the inverse of E = [[-1, 2], [4, -3]].
7. Find the inverse of the matrix G = [[5, 1], [2, 3]].
8. Verify if the matrix H = [[0.3, -0.1], [-0.2, 0.1]] is the inverse of G = [[5, 1], [2, 3]].
9. Find the inverse of the matrix I = [[-2, 1], [3, -4]].
10. Verify if the matrix J = [[-0.8, -0.2], [-0.6, -1.4]] is the inverse of I = [[-2, 1], [3, -4]].
