Reminder: an Elementary Matrix is an Identity matrix
on which a single row operation has been performed.

example: the elementary matrix A = is I _{2 x 2} with row 2 added to row 1.

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QUESTIONS

1) B =

a) Find 3 elementary matrices E ₁, E ₂, E ₃, such that E ₃ $ E ₂ $ E ₁ $ B = I.
b) Find 3 elementary matrices D ₁, D ₂, D ₃, such that D ₁ $ D ₂ $ D ₃ = B.

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2) C =

a) Find 4 elementary matrices E ₁, E ₂, E ₃, E ₄, such that E ₄ $ E ₃ $ E ₂ $ E ₁ $ C = I.
b) Find 4 elementary matrices D ₁, D ₂, D ₃, D ₄ such that D ₁ $ D ₂ $ D ₃ $ D ₄ = C.

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3) A = and B =

Find 2 elementary matrices E ₁ and E ₂ so that E ₂ $ E ₁ $ A = B

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4) A = and B =

Find 2 elementary matrices E ₁ and E ₂ so that E ₁ $ A = E ₂ $ B

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5) If A = use elementary matrices to find A ^{- 1} if it exists.

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SOLUTIONS

1) B =

a) The 3 row operations we want to perform on B to obtain I are:

i) row 2 minus 2(row 1)	ii) -1/3(row 2)	iii) row 1 minus 3(row 2)
E 1 = , so E 1 B =	E 2 = , so E 2 E 1 B =	E 3 = , so E 3 E 2 E 1 B =

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b) now we want to do the inverse of the 3 row operations in the reverse order*.
So the 3 row operations we want to perform on I to obtain B are:

iii) row 2 plus 2(row 1)	ii) - 3 (row 2)	i) row 1 plus 3(row 2)
D 1 = , so D 1 D 2 D3 = = B	D 2 = , so D 2 D3 =	D 3 = ,

* Note: we did these operations in the reverse order because of how the question is stated.
Had we been asked to find these matrices such that D ₃ $ D ₂ $ D ₁ = B, we
would have done the operations in the same order we used in part (a).
Remember that matrix multiplication on the right and on the left are not
necessarily equivalent operations.

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2) C =

a) We want to turn C into I so here's the 4 operations we should do:

i) switch row 3 and row 1 E 1 = , so E 1 C =	ii) take ½ (row 2) E 2 = , so E 2 E 1 C =
iii) row 3 minus 6(row 2) E 3 = , so E 3 E 2 E 1 C =	iv) row 2 minus 2(row 3) E 4 = , so E 4 E 3 E 2 E 1 C == I

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b) Just as in question 1, we want the inverse operations in the reverse order.

iv) switch row 3 and row 1 D 1 = ,	iii) take 2 (row 2) E 2 = ,
ii) row 3 plus 6(row 2) E 3 = ,	i) row 2 plus 2(row 3) E 4 = ,

** Note that the order is reversed. the 1st move is in the fourth table cell.

3) A = and B =

Since row 2 in both matrices is identical, we need to change row 1 and row 3.
To change Row 3 of A to Row 3 of B, we take row 3 minus 2(row 1).
This makes E ₁ = ,

Now, to change Row 1 of A into Row 1 of B, we take row 1 minus 4(row 2)

This makes E ₂ = and E ₂ $ E ₁ $ A = B

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4) A = and B =

It's pretty obvious that we have to switch the rows and double or halve a row.
The order we choose determines which set of answers we get.
i) If we double row 1 of A first, E ₁ = , and E ₁ A = ,
This will make E ₂ = , and E ₂ B = so that E ₁ $ A = E ₂ $ B

ii) If we flip the rows of A first, E ₁ = , and E ₁ A =

This will make E ₂ = , since we need to take ½ of row 2
E ₂ B = so that E ₁ $ A = E ₂ $ B

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5) We set up the "super-augmented" matrix with A on the left and I on the right,
then we do row reduction. When we have I on the left, we have A ^{- 1} on the right.

The steps which reduce A to I are:

i) switch R3 and R1

ii) R2 - 4(R1) R3 - 3(R1)

iii) - (R3 - R2)

iv) R1 - R3 R2 - 2(R3)

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