Question

A matrix A has x rows and x+5 columns. B has y rows and 11−y columns. Both AB and BA exist. Find x and y.

Answer 1

x = 3

y = 8

Explanation:

A matrix product exists if and only if the number of columns in the left matrix is equal to the number of rows in the right matrix.

Setup

For AB to exist, we must have ...

columns of A = rows of B

x +5 = y

For BA to exist, we must have ...

columns of B = rows of A

11 -y = x

Solution

Using the second equation to substitute for x in the first equation, we have ...

(11 -y) +5 = y

16 = 2y

8 = y

Then the value of x is ...

11 -8 = x = 3

The values of x and y are ...

• x = 3
• y = 8

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Additional comment

The dimensions are A = 3×8; B = 8×3.