For the product of two matrices to be defined, the number of columns in the first matrix must be equal to the number of rows in the second. This is the case, since has two columns and has two rows. is defined.

Matrix multiplication is worked by multiplying each row of the first matrix by each column of the second matrix. This is done by adding the products of the entries in corresponding positions. Thus,

The product of any scalar value and a matrix is the matrix formed when each entry in the matrix is multiplied by that scalar. Thus, if

,

then

The statement is false, since the entry in Row 3, Column 1 is incorrect.

First, it must be established that is defined. This is the case if and only and have the same number of rows, which is true, and they have the same number of columns, which is also true. is therefore defined.

Addition of two matrices is performed by adding corresponding elements together, so

The statement is true.

First, it must be established that  is defined. This is the case if and only and have the same number of rows, which is true, and they have the same number of columns, which is also true.  is therefore defined.

Subtraction of two matrices is performed by subtracting corresponding elements together, so

The statement is true.

First, make the y-coefficients each other's opposite. This can be done by multiplying the first equation by 2 on both sides:

The y-coefficients of the two equations are now opposites, so, if the left and right sides of the two equations are added, the y-terms will cancel out, as follows:

The resulting statement is identically false. It follows that the two equations of the system are inconsistent with each other. The system has no solution.

First, make the y-coefficients each other's opposite. This can be done by multiplying the first equation by 2 on both sides:

The y-coefficients of the two equations are now opposites, so, if the left and right sides of the two equations are added, the y-terms will cancel out, as follows:

The resulting statement is identically true. It follows that the two equations of the system form a dependent system. The solution set can be found by allowing to be arbitrary, and to solve for in either equation. Selecting the first:

The solution set is

, arbitrary.

The Leontief model for an input-output system requires the following steps.

Let be the technology matrix for the system, the entries of which are the amounts of each product consumed in the manufacture of each product. Letting the first row/column represent Food A and the second row/column represent Food B, this matrix is

Note that ounces have been converted to pounds.

Let be the external demand vector, which gives the amount of each product demanded by outside consumers:

The output vector , which gives the amount of each product that must be produced to meet the demand, can be obtained through the matrix equation

 The top entry is the number of pounds of Food A that must be produced - 2,571 pounds.

The Leontief model for an input-output system requires the following steps.

Let be the technology matrix for the system, the entries of which are the amounts of each product consumed in the manufacture of each product. Letting the first row/column represent Food A and the second row/column represent Food B, this matrix is

Note that ounces have been converted to pounds.

Let be the external demand vector, which gives the amount of each product demanded by outside consumers:

The output vector , which gives the amount of each product that must be produced to meet the demand, can be obtained through the matrix equation

 The bottom entry is the number of pounds of Food B that must be produced - 3,286 pounds.

To find , the inverse of , set up the augmented matrix as follows:

Apply the Gauss-Jordan elimination process to this matrix by performing row operations on it until the matrix is formed - that is,

.

Work each of the first three columns in left to right order until the entry in the diagonal position is a one and the other two entries are zeroes.

One possible set of row operations, with the resulting matrices, is as follows:

This is the desired result. The inverse is seen in the last three columns:

To find , the inverse of , set up the augmented matrix as follows:

Apply the Gauss-Jordan elimination process to this matrix by performing row operations on it until the matrix is formed - that is,

.

Work each of the first three columns, right-to-left, until the entry in the diagonal position is a one and the other two entries are zeroes.

One possible sequence of row operations, with the resulting matrices, is as follows:

This is the desired result. The inverse is seen in the last three columns: