## Halmos Sections 37 and 38: Matrices and Matrices of Linear Transformations(Pt. IV)

**Point of post: **This is a continuation of this post.

*Remark: *For some strange reason the fourth (this one) and the fifth (the previous one) got mixed up in the order of posting. The number is correct, this is the fourth post in this sequence and the one preceding it the fifth.

16.

**Problem: **Decide which of the following matrices are invertible and find the inverses of the one that are.

**a) **

**b) **

**c) **

**d) **

**e) **

**f) **

**g) **

**Proof:**

It’s clear that if is any matrix (in general) then if for some non-zero then can’t be invertible.

**a) **This is invertible, since serves as an inverse.

**b) **This is not invertible since

**c) **This isn’t since

**d) **This is since

**e) **This isn’t since

**f) **This isn’t since

**g) **This isn’t since

17.

**Problem: **For which values of are the following matrices invertible? Find the inverses whenever possible.

**a) **

**b) **

**c) **

**d) **

**Proof:**

**a) **This is always invertible since serves as an inverse.

**b) **This is invertible precisely when . Indeed, if then . But, if then functions as an inverse.

**c) **This is never invertible. Indeed, .

**d) **This is invertible precisely when . Indeed, if we’ve already proven in the last exercise that this matrix isn’t invertible, and if then serves as an inverse.

**References:**

1. Halmos, Paul R. *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print

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