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In Exercises l and 2, compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let.
In Exercises l and 2, compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let.
In the rest of this exercise set and in those to follow, you should assume that each matrix expression is defined. That is, the sizes of the matrices (and vectors) involved “match” appropriately.
Let
In the rest of this exercise set and in those to follow, you should assume that each matrix expression is defined. That is, the sizes of the matrices (and vectors) involved “match” appropriately.
Compute
In Exercises 5 and
In Exercises 5 and
Let
Let
Let
Let
products
Exercises 15 and 16 concern arbitrary matrices
a. If
b. Each column of
c.
d.
e. The transpose of a product of matrices equals the product of their transposes in the same order.
Exercises 15 and 16 concern arbitrary matrices
a. If
b. The second row of
c.
d.
e. The transpose of a sum of matrices equals the sum of their transposes.
Suppose the first two columns, by and
Suppose the third column of
Suppose the second column of
Suppose the last column of
Show that if the columns of
Suppose
Suppose
Suppose
Suppose
In Exercises 27 and
Let
In Exercises 27 and
If
Prove Theorem 2
Show that
Show that
Give a formula for
[M] Read the documentation for your matrix program, and write the commands that will produce the following matrices (without keying in each entry of the matrix).
a.
b.
c. The
d.
A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.
[M] Write the command(s) that will create a
A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.
[M] Construct a random
A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.
[M] Use at least three pairs of random
A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.
[M] Let
Compute
A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.
[M] Describe in words what happens when you compute