How To Find Inverse Of A Matrix METHOD 1:
Creating the Adjugate Matrix to Find the Inverse Matrix
- You have to calculate the determinant of the 3×3 matrix as the first step. If the founded determinant is 0, then your work is done here, because the matrix has no inverse. The determinant of matrix M can be represented by det(M).
- For a 3×3 matrix, find the determinant is the first step
2. Transpose the matrix. Transposing means swapping the (i,j)th element and the (j,i)th. When you transpose the terms of the matrix, you should see that the main diagonal (from upper left to lower right) is unaltered.
- Another way to think about the transposition of that you re-write the first line as the first column, the middle line becomes the middle column, the third row becomes the third column. Pay attention to the color elements in the diagram above, and see where the numbers changed the situation.
3. Find determinant of every 2×2 small matrices.Every element newly transposed matrix of 3×3 is connected to the corresponding 2×2 “minor” matrix.To find the right secondary matrix for each member, first select the row and column of the term, which begins with you. This should include the five members of the matrix. The remaining four members are a small matrix.
- In the above example, if you want a minor matrix of the term in the second row, first column, you allocate five terms that are in the second row and first column. The remaining four members are the corresponding minor of the matrix.
- Find the determinant of each small cross-multiplying diagonals and subtraction, as shown in Figure.
- For more on minor matrices and their uses, see Understand the Basics of Matrices.
4. Create a matrix of cofactors. Place the previous step results in a new matrix cofactor, aligning each secondary determinant of the matrix with a corresponding position in the original matrix. Thus, the determinant of which you expected from (1.1) of the original matrix is in the position (1,1). Then you have to change the sign of the alternating conditions of this new matrix, after the “checkerboard” image.
- When assigning attributes, the first element of the first row retains its original sign.The reversed. The second element the third element retains its original sign.Continue with the rest of the matrix in this fashion.
- Please note that the (+) or (-) signs on a checkerboard pattern do not suggest that the final deadline is to be positive or negative.They are accounting indicators (+) or backward (-) sign number was originally independently.
- For a review of cofactors, see Understand the Basics of Matrices.
The final result of this step is called the adjugate matrix of the original. This is sometimes referred to as the adjoint matrix. The adjugate matrix is noted as Adj(M).
5. Divide each term of the adjugate matrix by the determinant of the matrix. Recall the determinant of M that you calculated in the initial step (to check that the inverse was possible or not). You now divide every term of the matrix by that value of the determinant det(M). Place the result of each calculation into the spot of the original term. The result is the inverse of the original matrix.
- For the sample matrix shown in the diagram, the determinant is 1. Therefore, dividing every term of the adjugate matrix results in the adjugate matrix itself. (You won’t always be so lucky.)
- Instead of dividing, some sources represent this step as multiplying each term of M by 1/det(M). Mathematically, these are equivalent.
How To Find Inverse Of A Matrix METHOD 2:
Using Linear Row Reduction to Find the Inverse Matrix
- Attach the unit matrix to the initial matrix. Make a note of the original matrix M, draw a vertical line to the right of it, and then write the identity matrix to the right of it. You should now have what appears to be a matrix with three rows of six each colony.
- Recall that the identity matrix is a special matrix with 1s in each position of the principal diagonal from upper left to lower right, and 0s in all other positions. For an overview of the identity matrix, and its properties, see Understanding Fundamentals matrices.
2. The task of reducing operations.Your linear series is to create an identity matrix at the left side of this new augmented matrix. As you perform the steps of reducing lines on the left side, you have to consistently perform the same operation on the right, which began as a template.
- Note that several abbreviations are performed as a combination of scalar multiplication and addition or subtraction of lines in order to isolate individual member’s matrix. For a fuller review see Row-Reduce matrices.
3. Continue until obrazuesh matrix.Keep identically repeating linear operation to reduce the row to the left part of the expanded matrix displays the identity matrix (diagonal of 1s, with other terms 0). When you have reached this point, the right part of your vertical divider will be the inverse of the original matrix.
4. Write out the inverse matrix. Copy the elements now appearing on the right side of the vertical divider as the inverse matrix.