6.4. EXERCISES 125

34. Use the formula for the inverse in terms of the cofactor matrix to find if possible theinverses of the matrices(

1 11 2

),

 1 2 30 2 14 1 1

 ,

 1 2 12 3 00 1 2

 .

If the inverse does not exist, explain why.

35. Here is a matrix,  1 0 00 cos t −sin t0 sin t cos t

Does there exist a value of t for which this matrix fails to have an inverse? Explain.

36. Here is a matrix,  1 t t2

0 1 2tt 0 2

Does there exist a value of t for which this matrix fails to have an inverse? Explain.

37. Here is a matrix,  et cosh t sinh tet sinh t cosh tet cosh t sinh t

Does there exist a value of t for which this matrix fails to have an inverse? Explain.

38. Show that if det(A) ̸= 0 for A an n×n matrix, it follows that if Ax = 0, then x = 0.

39. Suppose A,B are n×n matrices and that AB = I. Show that then BA = I. Hint: Youmight do something like this: First explain why det(A) ,det(B) are both nonzero.Then (AB)A = A and then show BA(BA− I) = 0. From this use what is given toconclude A(BA− I) = 0. Then use Problem 38.

40. Use the formula for the inverse in terms of the cofactor matrix to find the inverse ofthe matrix

A =

 et 0 00 et cos t et sin t0 et cos t− et sin t et cos t + et sin t

 .

41. Find the inverse if it exists of the matrix et cos t sin tet −sin t cos tet −cos t −sin t

 .