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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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inverse of matrix [ 1 3 -2] [-3 0-5] [2 5 0] Related: Examples : Find...
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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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inverse of matrix [ 1 3 -2] [-3 0-5] [2 5 0] Related: Examples : Finding Inverse of 3x3 Matrices? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about inverse of matrix [ 1 3 -2] [-3 0-5] [2 5 0] Related: Examples : Finding Inverse of 3x3 Matrices? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for inverse of matrix [ 1 3 -2] [-3 0-5] [2 5 0] Related: Examples : Finding Inverse of 3x3 Matrices?.
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