Matrices - SAT Subject Test in Math I

The inverse of a 2 x 2 matrix , if it exists, is the matrix

First, we need to establish that the inverse is defined, which it is if and only if determinant .

Set , and check:

The determinant is equal to 0, so does not have an inverse.

When subtracting matrices, you want to subtract each corresponding cell.

Now solve for and

This problem involves a scalar multiplication with a matrix. Simply distribute the negative three and multiply this value with every number in the 2 by 3 matrix. The rows and columns will not change.

Matrix addition is very easy! All that you need to do is add each correlative member to each other. Think of it like this:

Now, just simplify:

There is your answer!

Matrix addition is really easy—don't overthink it! All you need to do is combine the two matrices in a one-to-one manner for each index:

Then, just simplify all of those simple additions and subtractions:

Scalar multiplication and addition of matrices are both very easy. Just like regular scalar values, you do multiplication first:

The addition of matrices is very easy. You merely need to add them directly together, correlating the spaces directly.

You can begin by treating this equation just like it was:

That is, you can divide both sides by :

Now, for scalar multiplication of matrices, you merely need to multiply the scalar by each component:

Then, simplify:

Therefore,

Begin by distributing the fraction through the matrix on the left side of the equation. This will simplify the contents, given that they are factors of :

Now, this means that your equation looks like:

This simply means:

and

or

Therefore,

You can treat matrices just like you treat other members of an equation. Therefore, you can subtract the matrix

from both sides of the equation. This gives you:

Now, matrix subtraction is simple. You merely subtract each element, matching the correlative spaces with each other:

Then, you simplify:

Therefore,

The dimensions of the matrices are 2 by 2.

The end result will also be a 2 by 2.

Evaluate the matrix.

The correct answer is:

The inverse of any two-by-two matrix can be found according to this pattern:

If

then

,

where determinant is equal to .

Therefore, if , then , the second row/first column entry in the matrix , can be found by setting , then evaluating:

.

A matrix lacks an inverse if and only if its determinant is equal to zero. The determinant of is

.

We seek the value of that sets this quantity equal to 0. Setting it as such then solving for :

,

the correct response.

A matrix lacks an inverse if and only if its determinant is equal to zero. The determinant of is

Setting this equal to 0 and solving for :

A matrix lacks an inverse if and only if its determinant is equal to zero. The determinant of is

, so

Since the square of all real numbers is nonnegative, this equation has no real solution. It follows that the determinant cannot be 0 for any real value of , and that must have an inverse for all real .

A matrix lacks an inverse if and only if its determinant is equal to zero. The determinant of is

Set this equal to 0 and solve for :

,

the correct response.

To compute the matrices, simply add the terms with the correct placement in the matrices. The resulting matrix is two by two.

The answer is: