Precalculus : Matrices and Vectors

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Find The Multiplicative Inverse Of A Matrix

What is the inverse of the following nxn matrix 

 

Possible Answers:

The matrix is not invertible.

Correct answer:

The matrix is not invertible.

Explanation:

Note the first and the last columns are equal.

Therefore, when we try to find the determinant using the following formula we get the determinant equaling 0:

This means simply, that the matrix does not have an inverse.

 

Example Question #1 : Find The Multiplicative Inverse Of A Matrix

Find the inverse of the matrix

.

Possible Answers:

Does not exist

Correct answer:

Does not exist

Explanation:

For a 2x2 matrix

the inverse can be found by 

Because the determinant is equal to zero in this problem, or

,

the inverse does not exist.

Example Question #3 : Inverses Of Matrices

Find the inverse of the matrix 

.

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

There are a couple of ways to do this. I will use the determinant method.

First we need to find the determinant of this matrix, which is

 

for a matrix in the form:

 .

Substituting in our values we find the determinant to be:

Now one formula for finding the inverse of the matrix is

 

.

Example Question #4 : Inverses Of Matrices

Find the inverse of the matrix.

Possible Answers:

Correct answer:

Explanation:

We use the inverse of a 2x2 matrix formula to determine the answer. Given a matrix 

 it's inverse is given by the formula:

First we define the determinant of our matrix:

Then, 

 

Example Question #5 : Inverses Of Matrices

Find the inverse of the following matrix.

Possible Answers:

This matrix has no inverse.

Correct answer:

This matrix has no inverse.

Explanation:

This matrix has no inverse because the columns are not linearly independent. This means if you row reduce to try to compute the inverse, one of the rows will have only zeros, which means there is no inverse.

Example Question #1 : Find The Multiplicative Inverse Of A Matrix

Find the multiplicative inverse of the following matrix:

 

Possible Answers:

This matrix has no inverse.

Correct answer:

Explanation:

By writing the augmented matrix , and reducing the left side to the identity matrix, we can implement the same operations onto the right side, and we arrive at , with the right side representing the inverse of the original matrix.

Example Question #7 : Inverses Of Matrices

What is the inverse of the identiy matrix   ?

Possible Answers:

The identity matrix 

Correct answer:

The identity matrix 

Explanation:

By definition, an inverse matrix is the matrix B that you would need to multiply matrix A by to get the identity. Since the identity matrix yields whatever matrix it is being multiplied by, the answer is the identity itself.

Example Question #1 : Find The Sum And Difference Of Vectors

Evaluate 

Possible Answers:

None of the other answers

Correct answer:

Explanation:

When adding two vectors, they need to be expanded into their components. Luckily, the problem statement gives us the vectors already in their component form. From here, we just need to remember that we can only add like components. So for this problem we get:

Now we can combine those values to write out the complete vector:

Example Question #1 : Find The Sum And Difference Of Vectors

What is the magnitude and angle for the following vector, measured CCW from the x-axis?

Possible Answers:

Correct answer:

Explanation:

The magnitude of the vector is found using the distance formula:

 

Vq02_02

To calculate the angle we must first find the inverse tangent of :

This angle value is the principal arctan, but it is in the fourth quadrant while our vector is in the second. We must add the angle 180° to this value to arrive at our final answer.

Example Question #1 : Find The Sum And Difference Of Vectors

Vector has a magnitude of 3.61 and a direction 124° CCW from the x-axis. Express  in unit vector form.

Possible Answers:

Correct answer:

Explanation:

For vector , the magnitude is doubled, but the direction remains the same.

Vq03_01

For our calculation, we use a magnitude of:

The x-coordinate is the magnitude times the cosine of the angle, while the y-coordinate is the magnitude times the sine of the angle.

The resultant vector is: .

Vq03_02

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