416 CHAPTER 19. EIGENVALUES AND EIGENVECTORS

By Theorem 19.2.20 this matrix has the same determinant as the original matrix. Remem-ber you can work with the columns also. Take −5 times the last column and add to thesecond column. This yields 

1 −8 3 20 0 −1 −10 −8 −4 10 10 −8 −4

By Theorem 19.2.22 this matrix has the same determinant as the original matrix. Now take(−1) times the third row and add to the top row. This gives.

1 0 7 10 0 −1 −10 −8 −4 10 10 −8 −4

which by Theorem 19.2.20 has the same determinant as the original matrix. Lets expandit now along the first column. This yields the following for the determinant of the originalmatrix.

det

 0 −1 −1−8 −4 110 −8 −4

which equals

8det(

−1 −1−8 −4

)+10det

(−1 −1−4 1

)=−82

I suggest you do not try to be fancy in using row operations. That is, stick mostly tothe one which replaces a row or column with a multiple of another row or column added toit. Also note there is no way to check your answer other than working the problem morethan one way. To be sure you have gotten it right you must do this. Unfortunately, thisprocess can go on and on when you keep getting different answers. This is a good exampleof something for which you should use a computer algebra system.

19.3 MATLAB and DeterminantsMATLAB can find determinants. Here is an example.

>> A=[1,3,2,4;-5,7,2,3;2,3,7,11;1,2,3,4]; det(A)Then press enter and you getans =-102.0000To enter a complex number 1+ 2i for example, you type: complex(1,2). However,

when MATLAB gives the answer, it will write it in the usual form 1+ 2i. If you havematrices in which there are complex entries, you can go ahead and let MATLAB do thetedious computations for you.

19.4 Applications19.4.1 A Formula for the InverseThe definition of the determinant in terms of Laplace expansion along a row or columnalso provides a way to give a formula for the inverse of a matrix. Recall the definition of