# SAT II Math II : Matrices

## Example Questions

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### Example Question #1 : Matrices

Multiply:

Explanation:

The product of a 2 x 2 matrix and a 2 x 1 matrix is a 2 x 1 matrix.

Multiply each row in the first matrix by the column matrix by multiplying elements in corresponding positions, then adding the products, as follows:

\

### Example Question #1 : Matrices

Multiply:

Explanation:

The product of a 2 x 2 matrix and a 2 x 1 matrix is a 2 x 1 matrix.

Multiply each row in the first matrix by the column matrix by multiplying elements in corresponding positions, then adding the products, as follows:

### Example Question #3 : Matrices

Define matrix

For which of the following matrix values of  is the expression  defined?

The expression  is defined for all of the values of  given in the other responses.

Explanation:

For the matrix product  to be defined, itis necessary and sufficent for the number of columns in  to be equal to the number of rows in

has two columns. Of the choices, only

has two rows, making it the correct choice.

### Example Question #1 : Matrices

Calculate:

Explanation:

To subtract two matrices, subtract the elements in corresponding positions:

### Example Question #1 : Matrices

Evaluate:

Explanation:

The determinant of the matrix  is

Substitute :

### Example Question #1 : Matrices

Give the determinant of the matrix

Explanation:

The determinant of the matrix  is

Substitute :

### Example Question #1 : Matrices

Multiply:

Explanation:

The product of a 2 x 2 matrix and a 2 x 1 matrix is a 2 x 1 matrix.

Multiply each row in the first matrix by the column matrix by multiplying elements in corresponding positions, then adding the products, as follows:

### Example Question #8 : Matrices

Let .

Give .

is not defined.

is not defined.

Explanation:

has three rows and two columns; since the number of rows is not equal to the number of columns,  is not a square matrix, and, therefore, it does not have an inverse.

### Example Question #9 : Matrices

Define matrix  .

For which of the following matrix values of  is the expression  defined?

I:

II:

III:

I, II, and III

I and II only

II and II only

I only

I and III only

I only

Explanation:

For the matrix sum  to be defined, it is necessary and sufficent for  and  to have the same number of rows and the same number of columns.  has three rows and two columns; of the three choices, only (I) has the same dimensions.

### Example Question #10 : Matrices

Let  and  be the 2 x 2 identity matrix.

Let .

Which of the following is equal to ?

Explanation:

The 2 x 2 identity matrix is  .

, or, equivalently,

,

so

Subtract the elements in the corresponding positions:

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