Take Note Property: Dimensions of a Product Matrix

If A is an m × n m times n  matrix and B is an n × p n times p  matrix, then the product matrix AB is an m × p m times p  matrix.

Product matrix AB is a 3 × 4 3 times 4  matrix.

Problem 5 Determining Whether Product Matrices Exist

Does either product AB or BA exist?

• A = [ − 2 1 3 − 2 0 1 ] B = [ − 1 0 2 1 2 0 0 3 ] eh equals . matrix with 3 rows and 2 columns , row1 column 1 , negative 2 , column 2 1 , row2 column 1 , 3 , column 2 negative 2 , row3 column 1 , 0 , column 2 1 , end matrix . b equals . matrix with 2 rows and 4 columns , row1 column 1 , negative 1 , column 2 0 , column 3 2 , column 4 1 , row2 column 1 , 2 , column 2 0 , column 3 0 , column 4 3 , end matrix

A B B A ( 3 × 2 ) ( 2 × 4 ) → 3 × 4  product matrix ( 2 × 4 ) ( 3 × 2 ) → no product ▲ ▲   ▲ ▲ equal Product  A B  exists . not equal table with 3 rows and 1 column , row1 column 1 , table with 1 row and 2 columns , row1 column 1 , eh b , column 2 b eh , end table , row2 column 1 , table with 1 row and 2 columns , row1 column 1 , open 3 times 2 close open 2 times 4 close rightwards arrow 3 times 4 . productmatrix , column 2 open 2 times 4 close open 3 times 2 close rightwards arrow . noproduct , end table , row3 column 1 , table with 2 rows and 3 columns , row1 column 1 , black up pointing triangle black up pointing triangle , column 2 , column 3 black up pointing triangle black up pointing triangle , row2 column 1 , table with 1 row and 1 column , row1 column 1 , equal , end table , column 2 cap product eh b , exists , . , column 3 table with 1 row and 1 column , row1 column 1 , notequal , end table , end table , end table

✓ Got It?

5. Do the following products exist?

   A = [ 1 4 − 3 5 ] B = [ − 1 1 ] C = [ 4 2 0 1 3 5 ] eh equals . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 4 , row2 column 1 , negative 3 , column 2 5 , end matrix b equals left bracket negative 1 1 right bracket . c equals . matrix with 2 rows and 3 columns , row1 column 1 , 4 , column 2 2 , column 3 0 , row2 column 1 , 1 , column 2 3 , column 3 5 , end matrix

   1. AB
   2. BA
   3. AC
   4. CA
   5. BC

Take Note Properties: Matrix Multiplication

If A, B, and C are n × n n times n  matrices, and O is the n × n n times n  zero matrix, then