a. Find a noninvertible matrix whose entries are four distinct prime numbers, or explain why no such matrix exists.
b. Find a noninvertible matrix whose entries are nine distinct prime numbers, or explain why no such matrix exists.
a. No such matrix exists.
Recall that for the general matrix's determinant is calculated with the formula:
Since a,b,c and d are distinct prime numbers, so
Therefore, such matrix exists.
A matrix is not invertible if one of it's columns is an linear combination of another two. We can arrange the two first columns to be linearly independent and we can attempt to find a linear combination of the two that will yield the third column. The only condition is that the new column is full of prime numbers. This can be arranged in numerous ways, one example is given below:
In the example given, the third column is equal to the sum of the second column multiplied by two and the first column.
In Exercises 5 through 40, find the matrix of the given linear transformation with respect to the given basis. If no basis is specified, use standard basis: for ,
for and for, . For the space of upper triangular matrices, use the basis
Unless another basis is given. In each case, determine whether is an isomorphism. If isn’t an isomorphism, find bases of the kernel and image of and thus determine the rank of .
21. from to with respect to the basis .
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