Solution to Quick Review

1. x + y

   

   Click on the picture for a three-dimensional look at the solution.

2. 

   

3. y-x

   

   Click on the picture for a three-dimensional look of the solution.

4. Ax i.e. times

   

   Note that in this multiplication, the values of the same subscript match, i.e.:

   

5. i.e. times

   

   As you can see, A is "too small" for y. There is nothing for to multiply by. Another way to think of this is that the size of A and y do not match, i.e.

   

   Unlike in question 4, where the subscript j in both A and x have the same values, the subscript i that appears in y and A in this question do not have the same values.

6. AB i.e. times

   

   In this question, a 2 x 3 matrix is multiplied to a 3 x 2 matrix. Consider the subscripts:

   

   Note that:

   • the inner matrix dimensions (pointed by arrows) agree
   • the outer matrix dimensions determine the size of the resulting matrix, i.e. the answer is a 2 x 2 matrix.

7. BA i.e. times

   

   In this question, a 3 x 2 matrix is multiplied to a 2 x 3 matrix. Consider the subscripts:

   

   Note that:

   • the inner matrix dimensions (pointed by arrows) agree
   • the outer matrix dimensions determine the size of the resulting matrix, i.e. the answer is a 3 x 3 matrix.

   Also note that . Usually, matrix multiplication is not commutative, as in this case. However, there are situations where two matrices are commutative, e.g.,

   

   In this case, XY = YX:

   

   Also note that if two matrices are inverse of each other, they are commutative as well.

8. AA i.e. times

   

   Here, a 2 x 3 matrix is multiplied to a 2 x 3 matrix. Looking at the matrix, you can see that there is nothing to be multiplied by -4. Consider the subscripts:

   

   Note that the inner matrix dimensions (pointed by arrows) do not agree. Therefore, the answer is undefined.