This isn't too hard, because we already calculated the determinants of the smaller parts when we did "Matrix of Minors". In practice we can just multiply each of the top row elements by the cofactor for the same location:. Is it the same? Which method do you prefer?
So it is often easier to use computers such as the Matrix Calculator. Then if you are left with a matrix with all zeros in a row, your matrix is not invertible. You do this by adding multiples of the first row as the "pivot row" to other rows, so that you get rid of the leading entries; in your matrix, start by adding -1 first row to the second row note that this is one of the three basic operations that does not change the solutions of your system. Add these three terms and you have found the determinant of A.
If so, then the matrix must be invertible. There are FAR easier ways to determine whether a matrix is invertible, however. If you have learned these methods, then here are two:. This can be done by inspection.
Do the same for the other two columns. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Asked 7 years, 5 months ago. You need to calculate the determinant of the matrix as an initial step. If the determinant is 0, then your work is finished, because the matrix has no inverse. The determinant of matrix M can be represented symbolically as det M. Transpose the original matrix. Transposing means reflecting the matrix about the main diagonal, or equivalently, 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 unchanged. Notice the colored elements in the diagram above and see where the numbers have changed position.
Find the determinant of each of the 2x2 minor matrices. To find the right minor matrix for each term, first highlight the row and column of the term you begin with. This should include five terms of the matrix. The remaining four terms make up the minor matrix. The remaining four terms are the corresponding minor matrix.
Find the determinant of each minor matrix by cross-multiplying the diagonals and subtracting, as shown. For more on minor matrices and their uses, see Understand the Basics of Matrices. Create the matrix of cofactors. Place the results of the previous step into a new matrix of cofactors by aligning each minor matrix determinant with the corresponding position in the original matrix. Thus, the determinant that you calculated from item 1,1 of the original matrix goes in position 1,1.
The second element is reversed. The third element keeps its original sign. Continue on with the rest of the matrix in this fashion. 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. Divide each term of the adjugate matrix by the determinant.
Recall the determinant of M that you calculated in the first step to check that the inverse was possible. You now divide every term of the matrix by that value. Place the result of each calculation into the spot of the original term. The result is the inverse of the original matrix. Therefore, dividing every term of the adjugate matrix results in the adjugate matrix itself. Mathematically, these are equivalent.
Method 2. Adjoin the identity matrix to the original matrix. Write out the original matrix M, draw a vertical line to the right of it, and then write the identity matrix to the right of that. You should now have what appears to be a matrix with three rows of six columns each. For a review of the identity matrix and its properties, see Understand the Basics of Matrices.
Perform linear row reduction operations. Your objective is to create the identity matrix on the left side of this newly augmented matrix.
As you perform row reduction steps on the left, you must consistently perform the same operations on the right, which began as your identity matrix. Remember that row reductions are performed as a combination of scalar multiplication and row addition or subtraction, in order to isolate individual terms of the matrix.
For a more complete review, see Row-Reduce Matrices. Continue until you form the identity matrix. Keep repeating linear row reduction operations until the left side of your augmented matrix displays the identity matrix diagonal of 1s, with other terms 0.
When you have reached this point, the right side of your vertical divider will be the inverse of your original matrix. Write out the inverse matrix. Copy the elements now appearing on the right side of the vertical divider as the inverse matrix.
Method 3. Select a calculator with matrix capabilities. Simple 4-function calculators will not be able to help you directly find the inverse. However, due to the repetitive nature of the calculations, an advanced graphing calculator, such as the Texas Instruments TI or TI, can greatly reduce the work.
Enter your matrix into the calculator. On the Texas Instruments calculators, you may need to press 2 nd Matrix. Select the Edit submenu. Select a name for your matrix. Most calculators are equipped to work with anywhere from 3 to 10 matrices, labeled with letters A through J. Typically, just choose [A] to work with. Hit the Enter key after making your selection.
Enter the dimensions of your matrix. This article is focusing on 3x3 matrices. However, the calculator can handle larger sizes. Enter the number of rows, then press Enter, and then the number of columns, and Enter.
Enter each element of the matrix. The calculator screen will show a matrix. If you previously were working with the matrix function, the prior matrix will appear on the screen. The cursor will highlight the first element of the matrix.
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