# SAT II Math I : Matrices

## Example Questions

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### Example Question #1 : Matrices

Define .

Give .

is not defined.

is not defined.

Explanation:

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.

### Example Question #2 : Matrices

Give the determinant of the matrix

Explanation:

The determinant of the matrix  is

Substitute :

### Example Question #16 : Matrices

Simplify:

Explanation:

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:

### Example Question #17 : Matrices

Simplify:

Explanation:

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:

### Example Question #42 : Matrices And Vectors

Evaluate:

Explanation:

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.

### Example Question #43 : Matrices And Vectors

Simplify:

Explanation:

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.

### Example Question #3 : Matrices

What is ?

Explanation:

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,

### Example Question #21 : Matrices

Given the following matrices, what is the product of  and ?

Explanation:

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

Now solve for  and

### Example Question #22 : Matrices

If , what is ?

Explanation:

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,

### Example Question #3 : Matrices

Simplify:

Explanation:

The dimensions of the matrices are 2 by 2.

The end result will also be a 2 by 2.

Evaluate the matrix.