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inverse of matrix [ 1 3 -2] [-3 0-5] [2 5 0] Related: Examples : Find...
Finding Inverse of 3x3 Matrices
To find the inverse of a 3x3 matrix, we can use the following formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A) \]
where det(A) is the determinant of matrix A and adj(A) is the adjugate of matrix A.

Step 1: Find the Determinant
- Calculate the determinant of matrix A using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Step 2: Find the Adjugate
- Find the cofactor matrix of matrix A by calculating the determinant of each 2x2 submatrix.
- Transpose the cofactor matrix to get the adjugate of matrix A.

Step 3: Calculate the Inverse
- Plug the determinant and adjugate into the formula to find the inverse of matrix A:
\[ A^{-1} = \frac{1}{\text{det}(A)} \times \text{adj}(A) \]
In the given example:
Matrix A = [1 3 -2; -3 0 -5; 2 5 0]

Step 1: Calculate the Determinant
\[ \text{det}(A) = 1(0 - (-5)) - 3(-5) + 2(15) = 5 + 15 + 30 = 50 \]

Step 2: Calculate the Adjugate
- Calculate the cofactor matrix and transpose it to get the adjugate of matrix A.

Step 3: Find the Inverse
- Plug the determinant and adjugate into the formula to find the inverse of matrix A.
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Introduction:To find the inverse of a matrix A using elementary transformations, we need to perform a series of row operations until A is transformed into the identity matrix I. Simultaneously, we perform the same row operations on the identity matrix I and obtain the inverse matrix A^-1.Given Matrix:A = [1 2 2 -1]Augmented Matrix:We will augment the given matrix A with the identity matrix I as follows:[A | I] = [1 2 2 -1 | 1 0 0 1]Row Operations:Perform the following row operations to transform A into I:1. R2 = R2 - 2R1[A | I] = [1 2 2 -1 | 1 0 0 1] [0 -2 -2 1 | -2 0 0 0] 2. R2 = -1/2R2[A | I] = [1 2 2 -1 | 1 0 0 1] [0 1 1/2 -1/2 | 1 0 0 0] 3. R1 = R1 - 2R2[A | I] = [1 0 1 -2 | -1 0 0 1] [0 1 1/2 -1/2 | 1 0 0 0] 4. R1 = R1 + R2[A | I] = [1 0 3/2 -5/2 | 0 0 0 1] [0 1 1/2 -1/2 | 1 0 0 0] Final Result:After performing the row operations, the matrix A is transformed into the identity matrix I. The inverse matrix A^-1 is given by the augmented matrix on the right side:A^-1 = [0 0 0 1 | 0 0 0 1] [0 1 1/2 -1/2 | 1 0 0 0]Explanation:By using elementary transformations, we performed a series of row operations on the given matrix A to transform it into the identity matrix I. Simultaneously, we performed the same row operations on the identity matrix I to obtain the inverse matrix A^-1. These row operations include adding or subtracting multiples of one row from another and multiplying a row by a constant. These operations ensure that the resulting matrix A^-1, when multiplied with the original matrix A, yields the identity matrix I. Therefore, A^-1 is the inverse of matrix A.

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inverse of matrix [ 1 3 -2] [-3 0-5] [2 5 0] Related: Examples : Finding Inverse of 3x3 Matrices?
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