Problem 1 Determining Whether Matrices are Inverses

For each of the following, are matrices A and B inverses?

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

   
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   Since A B = I eh b equals i  and B A = I , b eh equals i comma  matrices A and B are inverses.

2. A = [ 1 0 2 − 2 1 1 − 1 1 2 ] B = [ − 1 − 2 2 − 3 − 4 5 1 1 − 1 ] eh equals . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 0 , column 3 2 , row2 column 1 , negative 2 , column 2 1 , column 3 1 , row3 column 1 , negative 1 , column 2 1 , column 3 2 , end matrix . b equals . matrix with 3 rows and 3 columns , row1 column 1 , negative 1 , column 2 negative 2 , column 3 2 , row2 column 1 , negative 3 , column 2 negative 4 , column 3 5 , row3 column 1 , 1 , column 2 1 , column 3 negative 1 , end matrix

   
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   Since A B = I eh b equals i  and B A = I , b eh equals i comma  matrices A and B are inverses.

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

   
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   Since A B ≠ I eh b not equal to i  (and B A ≠ I b eh not equal to i  ), matrices A and B are not inverses.

✓ Got It?

1. For each of the following, are A and B inverses?
   1. A = [ 1 1 5 4 ] B = [ − 4 1 5 − 1 ] eh equals . matrix with 2 rows and 2 columns , row1 column 1 , 1 , column 2 1 , row2 column 1 , 5 , column 2 4 , end matrix b equals . matrix with 2 rows and 2 columns , row1 column 1 , negative 4 , column 2 1 , row2 column 1 , 5 , column 2 negative 1 , end matrix
   2. A = [ 3 2 5 4 ] B = [ 2 − 1 − 5 2 3 2 ] eh equals . matrix with 2 rows and 2 columns , row1 column 1 , 3 , column 2 2 , row2 column 1 , 5 , column 2 4 , end matrix b equals . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 negative 1 , row2 column 1 , negative , 5 halves , column 2 3 halves , end matrix
   3. Reasoning Does the matrix [ 0 0 0 0 ] . matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 0 , row2 column 1 , 0 , column 2 0 , end matrix  have an inverse? Explain.