- Here are some matrices:
- Here are some matrices:
- Here are some matrices:
^{T},3B − A^{T},AC,CA,AE,E^{T}B,BE,DE,EE^{T},E^{T}E. If it is not possible explain why. - Here are some matrices:
- Let A = , B =, and C =. Find if possible.
- AB
- BA
- AC
- CA
- CB
- BC

- 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}

- Let A = . Find all 2 × 2 matrices, B such that AB = 0.
- Let x =and y =. Find x
^{T}y and xy^{T}if possible. - Let A = ,B =. Is it possible to choose k such that AB = BA? If so, what should k equal?
- Let A = ,B =. Is it possible to choose k such that AB = BA? If so, what should k equal?
- In 18.15 - 18.22 describe −A and 0.
- Let A be an n × n matrix. Show A equals the sum of a symmetric and a skew
symmetric matrix. (M is skew symmetric if M = −M
^{T}. M is symmetric if M^{T}= M.) Hint: Show thatis symmetric and then consider using this as one of the matrices. - 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. - Suppose M is a 3 × 3 skew symmetric matrix. Show there exists a vector Ω such
that for all u ∈ ℝ
^{3}Hint: Explain why, since M is skew symmetric it is of the form

where the ω

_{i}are numbers. Then consider ω_{1}i + ω_{2}j + ω_{3}k. - Using only the properties 18.15 - 18.22 show −A is unique.
- Using only the properties 18.15 - 18.22 show 0 is unique.
- Using only the properties 18.15 - 18.22 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 18.15 - 18.22 and previous problems show A = −A.
- Prove 18.24.
- Prove that I
_{m}A = A where A is an m × n matrix. - Give an example of matrices, A,B,C such that B≠C, A≠0, and yet AB = AC.
- Suppose AB = AC and A is an invertible n × n matrix. Does it follow that B = C? Explain why or why not. What if A were a non invertible n × n matrix?
- Find your own examples:
- 2 × 2 matrices, A and B such that A≠0,B≠0 with AB≠BA.
- 2 × 2 matrices, A and B such that A≠0,B≠0, but AB = 0.
- 2 × 2 matrices, A, D, and C such that A≠0,C≠D, but AC = AD.

- Explain why if AB = AC and A
^{−1}exists, then B = C. - 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.
- Let A = .Find A
^{−1}if possible. If A^{−1}does not exist, determine why. - Let A = .Find A
^{−1}if possible. If A^{−1}does not exist, determine why. - Let A = . Find A
^{−1}if possible. If A^{−1}does not exist, determine why. - Let A = .Find A
^{−1}if possible. If A^{−1}does not exist, determine why. - Let A be a 2 × 2 matrix which has an inverse. Say A = . Find a formula for A
^{−1}in terms of a,b,c,d. - 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. - Write in the form Awhere A is an appropriate matrix.
- Write in the form Awhere A is an appropriate matrix.
- Write in the form Awhere A is an appropriate matrix.
- Using the inverse of the matrix, find the solution to the systems
- Using the inverse of the matrix, find the solution to the systems
- Using the inverse of the matrix, find the solution to the system
- 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. - Prove that if A
^{−1}exists and Ax = 0 then x = 0. - Show that if A
^{−1}exists for an n × n matrix, then it is unique. That is, if BA = I and AB = I, then B = A^{−1}. - Show that if A is an invertible n×n matrix, then so is A
^{T}and^{−1}=^{T}. - Show
^{−1}= B^{−1}A^{−1}by verifying that AB= I andHint: Use Problem 45.

- Show that
^{−1}= C^{−1}B^{−1}A^{−1}by verifying thatand

= I. Hint: Use Problem 45. - If A is invertible, show
^{−1}=^{2}. Hint: Use Problem 45. - If A is invertible, show
^{−1}= A. Hint: Use Problem 45. - 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}. In the notation above, Ax ⋅ y = x⋅A^{T}y. Use the definition of matrix multiplication to do this. - Use the result of Problem 51 to verify directly that
^{T}= B^{T}A^{T}without making any reference to subscripts. - Suppose A is an n × n matrix and for each j,
Show that the infinite series ∑

_{k=0}^{∞}A^{k}converges in the sense that the ij^{th}entry of the partial sums converge for each ij. Hint: Let R ≡ max_{j}∑_{i=1}^{n}. Thus R < 1. Show that≤ R^{2}. Then generalize to show that≤ R^{m}. Use this to show that the ij^{th}entry of the partial sums is a Cauchy sequence. From calculus, these converge by completeness of the real or complex numbers. Next show that^{−1}= ∑_{k=0}^{∞}A^{k}. The Leontief model in economics involves solving an equation for x of the formThe vector Ax is called the intermediate demand and the vectors A

^{k}x have economic meaning. From the above,The series is also called the Neuman series. It is important in functional analysis.

- An elementary matrix is one which results from doing a row operation to
the identity matrix. Thus the elementary matrix E which results from
adding a times the i
^{th}row to the j^{th}row would have aδ_{ik}+ δ_{jk}as the jk^{th}entry and all other rows would be unchanged. That is δ_{rs}provided r≠j. Show that multiplying this matrix on the left of an appropriate sized matrix A results in doing the row operation to the matrix A. You might also want to verify that the other elementary matrices have the same effect, doing the row operation which resulted in the elementary matrix to A. - Let a be a fixed vector. The function T
_{a}defined by T_{a}v = a + v has the effect of translating all vectors by adding a. Show this is not a linear transformation. Explain why it is not possible to realize T_{a}in ℝ^{3}by multiplying by a 3 × 3 matrix. - In spite of Problem 55 we can represent both translations and rotations by matrix
multiplication at the expense of using higher dimensions. This is done by the
homogeneous coordinates. I will illustrate in ℝ
^{3}where most interest in this is found. For each vector v =^{T}, consider the vector in ℝ^{4}^{T}. What happens when you doDescribe how to consider both rotations and translations all at once by forming appropriate 4 × 4 matrices.

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