HOW TO FIND THE DIMENSIONS OF A MATRIX WHEN MULTIPLYING

Determine whether each matrix product is defined. If so, state the dimensions of the product.

Problem 1 :

A_{3 × 5} · B_{5 × 2}

Solution:

The inner dimensions are equal so the matrix product is defined. The dimensions of the product are 3 × 2.

Problem 2 :

Solution:

The inner dimensions are not equal, so the matrix product is not defined.

Find each product, if possible.

Problem 3 :

Solution:

• Dimension of the first matrix = 1 x 2
• Dimension of the second matrix = 2 x 2
• Then the result matrix will have the dimension of 1 x 2.

Problem 4 :

Solution:

• Dimension of the first matrix = 2 x 1
• Dimension of the second matrix = 1 x 3
• Then the result matrix will have the dimension of 2 x 3.
  

Problem 5 :

Solution:

• The dimension of the first matrix is 2 x 3
• The dimension of the second matrix is 2 x 2

Since the number of columns and number of rows in the matrices are not same, the product cannot be found. So, it is not possible.

Problem 6 :

Solution:

• Dimension of the first matrix = 2 x 2
• Dimension of the second matrix = 2 x 1
• Then the result matrix will have the dimension of 2 x 1

Determine whether each matrix product is defined. If so, state the dimensions of the product.

Problem 7 :

A_{4 × 3} ⋅ B_{3 × 2}

Solution:

The inner dimensions are equal so the matrix product is defined. The dimensions of the product are 4 × 2.

Problem 8 :

X_{2 × 2} ⋅ Y_{2 × 2}

Solution:

The inner dimensions are equal so the matrix product is defined. The dimensions of the product are 2 × 2.

Problem 9 :

P_{1 × 3} ⋅ Q_{4 × 1}

Solution:

The inner dimensions are not equal, so the matrix product is not defined.

Problem 10 :

R_{1 × 4} ⋅ S_{4 × 5}

Solution:

The inner dimensions are equal so the matrix product is defined. The dimensions of the product are 1 × 5.

Problem 11 :

M_{4 × 3} ⋅ S_{4 × 3}

Solution:

The inner dimensions are not equal, so the matrix product is not defined.

Problem 12 :

A_{3 × 1} ⋅ B_{1 × 5}

Solution:

The inner dimensions are equal so the matrix product is defined. The dimensions of the product are 3 × 5.