Calculus 3 : Matrices

Example Questions

Example Question #31 : Matrices

Find the determinant of the matrix

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

For the matrix

The determinant is thus:

Example Question #32 : Matrices

Find the determinant of the matrix

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

For the matrix

The determinant is thus:

Example Question #33 : Matrices

Find the determinant of the matrix

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

For the matrix

The determinant is thus:

Example Question #34 : Matrices

Find the matrix product of , where  and

Explanation:

In order to multiply two matrices, , the respective dimensions of each must be of the form  and  to create an  (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that  and

The resulting matrix product is then:

Example Question #35 : Matrices

Find the matrix product of , where  and

Explanation:

In order to multiply two matrices, , the respective dimensions of each must be of the form  and  to create an  (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that  and

The resulting matrix product is then:

Example Question #36 : Matrices

Find the matrix product of , where  and

Explanation:

In order to multiply two matrices, , the respective dimensions of each must be of the form  and  to create an  (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that  and

The resulting matrix product is then:

Example Question #37 : Matrices

Find the matrix product of , where  and

Explanation:

In order to multiply two matrices, , the respective dimensions of each must be of the form  and  to create an  (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that  and

The resulting matrix product is then:

Example Question #38 : Matrices

Find the matrix product of , where  and

Explanation:

In order to multiply two matrices, , the respective dimensions of each must be of the form  and  to create an  (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that  and

The resulting matrix product is then:

Example Question #39 : Matrices

Find the matrix product of , where  and

Explanation:

In order to multiply two matrices, , the respective dimensions of each must be of the form  and  to create an  (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that  and

The resulting matrix product is then:

Example Question #40 : Matrices

Find the determinant of the matrix

Explanation:

The determinant of a 3x3 matrix can be found via a means of reduction into three 2x2 matrices as follows:

These can then be further reduced via the method of finding the determinant of a 2x2 matrix:

For the matrix

The determinant is thus:

(It is of note that if a matrix has a zero determinant, then its columns are linearly dependent. In other words, one column could be created by some via some combination of the other two.

Note how if you multiply the second column by two and then subtract the first column, the third column results.)