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
.