- In 4.1 - 4.8 describe −A and 0.
- Let A be an n × n matrix. Show A equals the sum of a symmetric and a skew symmetric matrix.
- Show every skew symmetric matrix has all zeros down the main diagonal. The main diagonal
consists of every entry of the matrix which is of the form a
_{ii}. It runs from the upper left down to the lower right. - We used the fact that the columns of a matrix A are independent if and only if Ax = 0 has only the zero solution for x. Why is this so?
- If A is m × n where n > m, explain why there exists x ∈ F
^{n}such that Ax = 0 but x≠0. - Using only the properties 4.1 - 4.8 show −A is unique.
- Using only the properties 4.1 - 4.8 show 0 is unique.
- Using only the properties 4.1 - 4.8 show 0A = 0. Here the 0 on the left is the scalar 0 and the 0 on the right is the zero for m × n matrices.
- Using only the properties 4.1 - 4.8 and previous problems show A = −A.
- Prove that I
_{m}A = A where A is an m × n matrix. - Let A and be a real m×n matrix and let x ∈ ℝ
^{n}and y ∈ ℝ^{m}. Show_{ℝm}=_{ℝn}where_{ℝk}denotes the dot product in ℝ^{k}. You need to know about the dot product. It will be discussed later but hopefully it has been seen in physics or calculus. - Use the result of Problem 11 to verify directly that
^{T}= B^{T}A^{T}without making any reference to subscripts. However, note that the treatment in the chapter did not depend on a dot product. - Let x =and y =. Find x
^{T}y and xy^{T}if possible. - Give an example of matrices, A,B,C such that B≠C, A≠0, and yet AB = AC.
- Let A = , B =, and C =. Find if possible the following products. AB,BA,AC,CA,CB,BC.
- Show
^{−1}= B^{−1}A^{−1}. - Show that if A is an invertible n × n matrix, then so is A
^{T}and^{−1}=^{T}. - Show that if A is an n × n invertible matrix and x is a n × 1 matrix such that Ax = b for b
an n × 1 matrix, then x = A
^{−1}b. - Give an example of a matrix A such that A
^{2}= I and yet A≠I and A≠ − I. - Give an example of matrices, A,B such that neither A nor B equals zero and yet AB = 0.
- Give another example other than the one given in this section of two square matrices, A and B such that AB≠BA.
- Suppose A and B are square matrices of the same size. Which of the following are correct?
^{2}= A^{2}− 2AB + B^{2}^{2}= A^{2}B^{2}^{2}= A^{2}+ 2AB + B^{2}^{2}= A^{2}+ AB + BA + B^{2}- A
^{2}B^{2}= AB ^{3}= A^{3}+ 3A^{2}B + 3AB^{2}+ B^{3}- = A
^{2}− B^{2} - None of the above. They are all wrong.
- All of the above. They are all right.

- Let A = . Find all 2 × 2 matrices, B such that AB = 0.
- Prove that if A
^{−1}exists and Ax = 0 then x = 0. - Let
Find A

^{−1}if possible. If A^{−1}does not exist, determine why. - Let
Find A

^{−1}if possible. If A^{−1}does not exist, determine why. - Let
Find A

^{−1}if possible. If A^{−1}does not exist, determine why. - Let
Find A

^{−1}if possible. If A^{−1}does not exist, determine why. - Let
Find A

^{−1}if possible. If A^{−1}does not exist, determine why. Do this in ℚ^{2}and in ℤ_{5}^{2}. - Let
Find A

^{−1}if possible. If A^{−1}does not exist, determine why. Do this in ℚ^{2}and in ℤ_{3}^{2}. - If you have any system of equations Ax = b, let ker≡. Show that all solutions of the system Ax = b are in ker+ y
_{p}where Ay_{p}= b. This means that every solution of this last equation is of the form y_{p}+ z where Az = 0. - Write the solution set of the following system as the span of vectors and find a basis for the solution
space of the following system.
- Using Problem 32 find the general solution to the following linear system.
- Write the solution set of the following system as the span of vectors and find a basis for the solution
space of the following system.
- Using Problem 34 find the general solution to the following linear system.
- Write the solution set of the following system as the span of vectors and find a basis for the solution
space of the following system.
- Using Problem 36 find the general solution to the following linear system.
- Show that 4.20 is valid for p = −1 if and only if each block has an inverse and that this condition holds if and only if A is invertible.
- Let A be an n×n matrix and let P
^{ij}be the permutation matrix which switches the i^{th}and j^{th}rows of the identity. Show that P^{ij}AP^{ij}produces a matrix which is similar to A which switches the i^{th}and j^{th}entries on the main diagonal. - You could define column operations by analogy to row operations. That is, you switch two columns, multiply a column by a nonzero scalar, or add a scalar multiple of a column to another column. Let E be one of these column operations applied to the identity matrix. Show that AE produces the column operation on A which was used to define E.
- Consider the symmetric 3 × 3 matrices, those for which A = A
^{T}. Show that with respect to the usual notions of addition and scalar multiplication this is a vector space of dimension 6. What is the dimension of the set of skew symmetric matrices? - You have an m × n matrix of rank r. Explain why if you delete a column, the resulting matrix has rank r or rank r − 1.

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