In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix. The adjugate has sometimes been called the . The Adjoint of 3×3 Matrix block computes the adjoint matrix for the input matrix. Calculating the inverse of a 3×3 matrix by hand is a tedious job, but worth reviewing. You can also find the This is sometimes referred to as the adjoint matrix.

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Inverse operations are commonly used in algebra to simplify what otherwise might be difficult. For example, if a problem requires you to divide by a fraction, you can more easily multiply by its reciprocal. This is an inverse operation.

Similarly, since there is no division operator for matrices, you need to multiply by the inverse matrix. Calculating the inverse of a 3×3 matrix by hand is a tedious job, but worth reviewing.

You can also find the inverse using an advanced graphing calculator. To find the inverse of a 3×3 matrix, first calculate the determinant of the mztrix.

Inverse of a Matrix using Minors, Cofactors and Adjugate

If the determinant is 0, the matrix has no inverse. Next, transpose the matrix by rewriting the first row as the first column, the middle row as the middle column, and the third row as the third column. Find the determinant of each of the 2×2 minor matrices, then create a matrix of cofactors using the results of the previous step.

Divide each term of the adjugate matrix by the determinant to get the inverse.

A wikiHow Staff Editor reviewed this article to make sure it’s helpful and accurate. This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness. Together, they cited information from 18 referenceswhich can be found at the bottom of the page. This article has over 2, views, and 24 testimonials from our readers, earning it our reader-approved status. Check the determinant of the matrix.

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.

Adjugate matrix

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 adjointt 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 2×2 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 matdix referred to as the adjoint matrix. The adjugate matrix is noted as Adj M.

Ov 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. 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. Matrox 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. 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 mtarix 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. 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 adjlint calculator, such as the Texas Instruments TI or TI, can greatly reduce the work. Enter your matrix into the calculator. On the Admoint 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 3×3 arjoint.

However, the ov can handle larger sizes. Enter the number of rows, then press Enter, and then the number aadjoint columns, and Enter. Enter each element of the matrix.

The calculator screen will show aduoint 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. Type in the value of the matrix you wish to solve, and then Enter.

The cursor will move automatically to the next element of the matrix, overwriting any previous numbers. The matrix function will not read the number properly. Quit the Matrix function.

Compute adjoint of matrix – Simulink

After you have entered all values of the matrix, press the Quit key or 2 nd Quit, if necessary. This will exit you from the Matrix function and return you to the main display screen of your calculator. Use the inverse key to find the inverse matrix. First, reopen the Matrix function and use the Names button to select the matrix label that you used to define adjoiint matrix probably [A].

This may adjoimt using the 2 nd button, depending on your calculator. Press Enter, adjoimt the inverse matrix should appear on your screen. The calculator will not understand this operation.

If you receive an error message when you enter the inverse key, chances are that your original matrix does not have an inverse.

You may want to go back and calculate the determinant to find out. Convert your inverse matrix adoint exact answers. The first calculation that the calculator will give you is in decimal form. You should convert the decimal answers to fractional form, as necessary. If you are very lucky, all your results will be integers, but this is rare. The decimals will automatically appear as fractions. Find the determinant, then determine the co-factor matrix. Find the adj of the co-factor matrix, then divide through each term by the determinant.

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