Linear Algebra, Exploration

This subsection is optional. It briefly describes how an investigator might come to a good general definition, which is given in the next subsection.

The three cases above don't
show an evident pattern to use for the general n \ × \ n formula.
We may spot that the {}1 \ × \ 1 term
a has one letter, that the {}2 \ × \ 2 terms
a d and b c have two letters, and that the {}3 \ × \ 3
terms a e i, etc., have three letters.
We may also observe that in those terms
there is a letter from each row and column of the matrix, e.g.,
the letters in the c d h term

( \begin{matrix} & & c \\ d \\ & h\end{matrix} )

come one from each row and one from each column.
But these observations perhaps seem more puzzling than
enlightening.
For instance, we might wonder why
some of the terms are added while others are subtracted.

A good problem solving strategy is to
see what properties a solution must have and
then search for something with those properties.
So we shall start by asking what properties we require of the formulas.

At this point,
our primary way to decide whether a matrix is singular is to do Gaussian
reduction and then check whether
the diagonal of resulting echelon form matrix has any zeroes
(that is, to check whether the product down the diagonal is zero).
So, we may expect that the proof that a formula determines singularity
will involve applying Gauss' method to the matrix,
to show that in the end the product down the diagonal is zero if and only if
the determinant formula gives zero.
This suggests our initial plan: we will look for a family of
functions with the property of being
unaffected by row operations and with the property that a determinant of an
echelon form matrix is the product of its diagonal entries.
Under this plan, a proof that the functions determine singularity would go,
"Where T → ⋯ → \overset{̂}{T} is the Gaussian
reduction, the determinant of T equals the
determinant of \overset{̂}{T} (because the determinant is unchanged by row
operations), which is the product down the diagonal, which is
zero if and only if the matrix is singular".
In the rest of this subsection we will test this plan on the
{}2 \ × \ 2 and {}3 \ × \ 3 determinants that we know.
We will end up modifying the "unaffected by row operations"
part, but not by much.

The first step in checking the plan is to test whether
the {}2 \ × \ 2 and {}3 \ × \ 3 formulas are unaffected by the
row operation of pivoting: if

T {→}_{ \ }^{k {ρ}_{i} + {ρ}_{j}} \overset{̂}{T}

then is det ⁡ ( \overset{̂}{T} ) = det ⁡ ( T )?
This check of the {}2 \ × \ 2 determinant after the k {ρ}_{1} + {ρ}_{2} operation

det ⁡ ( ( \begin{matrix}a & b \\ k a + c & k b + d\end{matrix} ) ) = a ( k b + d ) − ( k a + c ) b = a d − b c

shows that it is indeed unchanged, and
the other {}2 \ × \ 2 pivot k {ρ}_{2} + {ρ}_{1} gives the same result.
The {}3 \ × \ 3 pivot k {ρ}_{3} + {ρ}_{2} leaves the determinant unchanged

\begin{matrix}det ⁡ ( ( \begin{matrix}a & b & c \\ k g + d & k h + e & k i + f \\ g & h & i\end{matrix} ) ) & = \begin{matrix}a ( k h + e ) i + b ( k i + f ) g + c ( k g + d ) h \\ \text{ } − h ( k i + f ) a − i ( k g + d ) b − g ( k h + e ) c\end{matrix} \\ & = a e i + b f g + c d h − h f a − i d b − g e c\end{matrix}

as do the other {}3 \ × \ 3 pivot operations.

So there seems to be promise in the plan.
Of course, perhaps
the {}4 \ × \ 4 determinant formula is affected by pivoting.
We are exploring a possibility here and we do not yet have all the facts.
Nonetheless, so far, so good.

The next step is to compare det ⁡ ( \overset{̂}{T} ) with
det ⁡ ( T ) for the operation

T {→}_{ \ }^{{ρ}_{i} ↔ {ρ}_{j}} \overset{̂}{T}

of swapping two rows.
The {}2 \ × \ 2 row swap {ρ}_{1} ↔ {ρ}_{2}

det ⁡ ( ( \begin{matrix}c & d \\ a & b\end{matrix} ) ) = c b − a d

does not yield a d − b c.
This {ρ}_{1} ↔ {ρ}_{3} swap inside of a {}3 \ × \ 3 matrix

det ⁡ ( ( \begin{matrix}g & h & i \\ d & e & f \\ a & b & c\end{matrix} ) ) = g e c + h f a + i d b − b f g − c d h − a e i

also does not give the same determinant as before the swap — again
there is a sign change.
Trying a different {}3 \ × \ 3 swap {ρ}_{1} ↔ {ρ}_{2}

det ⁡ ( ( \begin{matrix}d & e & f \\ a & b & c \\ g & h & i\end{matrix} ) ) = d b i + e c g + f a h − h c d − i a e − g b f

also gives a change of sign.

Thus, row swaps appear
to change the sign of a determinant.
This modifies our plan, but does not wreck it.
We intend to decide nonsingularity by considering
only whether the determinant is zero, not by considering its sign.
Therefore, instead of expecting determinants to be
entirely unaffected by row operations, will look for them to change sign
on a swap.

To finish,
we compare det ⁡ ( \overset{̂}{T} ) to det ⁡ ( T ) for the operation

T {→}_{ \ }^{k {ρ}_{i}} \overset{̂}{T}

of multiplying a row by a scalar k ≠ 0.
One of the {}2 \ × \ 2 cases is

det ⁡ ( ( \begin{matrix}a & b \\ k c & k d\end{matrix} ) ) = a ( k d ) − ( k c ) b = k ⋅ ( a d − b c )

and the other case has the same result.
Here is one {}3 \ × \ 3 case

\begin{matrix}det ⁡ ( ( \begin{matrix}a & b & c \\ d & e & f \\ k g & k h & k i\end{matrix} ) ) & = \begin{matrix}a e ( k i ) + b f ( k g ) + c d ( k h ) \\ \ − ( k h ) f a − ( k i ) d b − ( k g ) e c\end{matrix} \\ & = k ⋅ ( a e i + b f g + c d h − h f a − i d b − g e c )\end{matrix}

and the other two are similar.
These lead us to suspect that multiplying a row by k
multiplies the determinant by k.
This fits
with our modified plan because we are asking only that the
zeroness of the determinant be unchanged and we are not focusing on the
determinant's sign or magnitude.

In summary,
to develop the scheme for the formulas to compute determinants,
we look for determinant functions that remain unchanged
under the pivoting operation, that change sign on
a row swap, and that rescale on the rescaling of a row.
In the next two subsections we will find that for each n
such a function exists and is unique.

For the next subsection, note that, as above, scalars come out
of each row without affecting other rows.
For instance, in this equality

det ⁡ ( ( \begin{matrix}3 & 3 & 9 \\ 2 & 1 & 1 \\ 5 & 10 & − 5\end{matrix} ) ) = 3 ⋅ det ⁡ ( ( \begin{matrix}1 & 1 & 3 \\ 2 & 1 & 1 \\ 5 & 10 & − 5\end{matrix} ) )

the 3 isn't factored out of all three rows, only out of the top row.
The determinant acts on each row independently of the
other rows.
When we want to use this property of determinants, we shall
write the determinant as a function of the rows:
"det ⁡ ( {\overset{→}{ρ}}_{1} , {\overset{→}{ρ}}_{2} , … {\overset{→}{ρ}}_{n} )", instead of
as "det ⁡ ( T )"
or "det ⁡ ( {t}_{1 , 1} , … , {t}_{n , n} )".
The definition of the determinant that starts the next subsection is written
in this way.

Exercises

This exercise is recommended for all readers.

Problem 1

Evaluate the determinant of each.

• ( \begin{matrix}3 & 1 \\ − 1 & 1\end{matrix} )
• ( \begin{matrix}2 & 0 & 1 \\ 3 & 1 & 1 \\ − 1 & 0 & 1\end{matrix} )
• ( \begin{matrix}4 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 3 & − 1\end{matrix} )

Problem 2

Evaluate the determinant of each.

• ( \begin{matrix}2 & 0 \\ − 1 & 3\end{matrix} )
• ( \begin{matrix}2 & 1 & 1 \\ 0 & 5 & − 2 \\ 1 & − 3 & 4\end{matrix} )
• ( \begin{matrix}2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 1\end{matrix} )

This exercise is recommended for all readers.

Problem 3

Verify that the determinant of an upper-triangular
{}3 \ × \ 3 matrix is the product down the diagonal.

det ⁡ ( ( \begin{matrix}a & b & c \\ 0 & e & f \\ 0 & 0 & i\end{matrix} ) ) = a e i

Do lower-triangular matrices work the same way?

This exercise is recommended for all readers.

Problem 4

Use the determinant to decide if each is singular or
nonsingular.

• ( \begin{matrix}2 & 1 \\ 3 & 1\end{matrix} )
• ( \begin{matrix}0 & 1 \\ 1 & − 1\end{matrix} )
• ( \begin{matrix}4 & 2 \\ 2 & 1\end{matrix} )

Problem 5

Singular or nonsingular?
Use the determinant to decide.

• ( \begin{matrix}2 & 1 & 1 \\ 3 & 2 & 2 \\ 0 & 1 & 4\end{matrix} )
• ( \begin{matrix}1 & 0 & 1 \\ 2 & 1 & 1 \\ 4 & 1 & 3\end{matrix} )
• ( \begin{matrix}2 & 1 & 0 \\ 3 & − 2 & 0 \\ 1 & 0 & 0\end{matrix} )

This exercise is recommended for all readers.

Problem 6

Each pair of matrices differ by one row operation.
Use this operation to
compare det ⁡ ( A ) with det ⁡ ( B ).

• A = ( \begin{matrix}1 & 2 \\ 2 & 3\end{matrix} )

B = ( \begin{matrix}1 & 2 \\ 0 & − 1\end{matrix} )

• A = ( \begin{matrix}3 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 2\end{matrix} )

B = ( \begin{matrix}3 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1\end{matrix} )

• A = ( \begin{matrix}1 & − 1 & 3 \\ 2 & 2 & − 6 \\ 1 & 0 & 4\end{matrix} )

B = ( \begin{matrix}1 & − 1 & 3 \\ 1 & 1 & − 3 \\ 1 & 0 & 4\end{matrix} )

Problem 7

Show this.

det ⁡ ( ( \begin{matrix}1 & 1 & 1 \\ a & b & c \\ {a}^{2} & {b}^{2} & {c}^{2}\end{matrix} ) ) = ( b − a ) ( c − a ) ( c − b )

This exercise is recommended for all readers.

Problem 8

Which real numbers x make this matrix singular?

( \begin{matrix}12 − x & 4 \\ 8 & 8 − x\end{matrix} )

Problem 9

Do the Gaussian reduction to check
the formula for {}3 \ × \ 3 matrices stated in the preamble to
this section.

( \begin{matrix}a & b & c \\ d & e & f \\ g & h & i\end{matrix} )

is nonsingular iff
a e i + b f g + c d h − h f a − i d b − g e c ≠ 0

Problem 10

Show that the equation of a line in {ℝ}^{2} thru ( {x}_{1} , {y}_{1} )
and ( {x}_{2} , {y}_{2} ) is expressed by this determinant.

det ⁡ ( ( \begin{matrix}x & y & 1 \\ {x}_{1} & {y}_{1} & 1 \\ {x}_{2} & {y}_{2} & 1\end{matrix} ) ) = 0 \ {x}_{1} ≠ {x}_{2}

This exercise is recommended for all readers.

Problem 11

Many people know this mnemonic for the determinant of a {}3 \ × \ 3
matrix: first repeat the first two columns and then sum the products on the
forward diagonals and subtract the products on the backward diagonals.
That is, first write

( \begin{matrix}{h}_{1 , 1} & {h}_{1 , 2} & {h}_{1 , 3} & {h}_{1 , 1} & {h}_{1 , 2} \\ {h}_{2 , 1} & {h}_{2 , 2} & {h}_{2 , 3} & {h}_{2 , 1} & {h}_{2 , 2} \\ {h}_{3 , 1} & {h}_{3 , 2} & {h}_{3 , 3} & {h}_{3 , 1} & {h}_{3 , 2}\end{matrix} )

and then calculate this.

\begin{matrix}{h}_{1 , 1} {h}_{2 , 2} {h}_{3 , 3} + {h}_{1 , 2} {h}_{2 , 3} {h}_{3 , 1} + {h}_{1 , 3} {h}_{2 , 1} {h}_{3 , 2} \\ \ − {h}_{3 , 1} {h}_{2 , 2} {h}_{1 , 3} − {h}_{3 , 2} {h}_{2 , 3} {h}_{1 , 1} − {h}_{3 , 3} {h}_{2 , 1} {h}_{1 , 2}\end{matrix}

• Check that this agrees with the formula given in the
  preamble to this section.
• Does it extend to other-sized determinants?

Problem 12

The cross product of the vectors

\overset{→}{x} = ( \begin{matrix}{x}_{1} \\ {x}_{2} \\ {x}_{3}\end{matrix} ) \ \overset{→}{y} = ( \begin{matrix}{y}_{1} \\ {y}_{2} \\ {y}_{3}\end{matrix} )

is the vector computed as this determinant.

\overset{→}{x} × \overset{→}{y} = det ⁡ ( ( \begin{matrix}{\overset{→}{e}}_{1} & {\overset{→}{e}}_{2} & {\overset{→}{e}}_{3} \\ {x}_{1} & {x}_{2} & {x}_{3} \\ {y}_{1} & {y}_{2} & {y}_{3}\end{matrix} ) )

Note that the first row is composed of vectors, the vectors from the
standard basis for {ℝ}^{3}.
Show that the cross product of two vectors is perpendicular to each vector.

Problem 13

Prove that each statement holds for {}2 \ × \ 2 matrices.

• The determinant of a
  product is the product of the determinants
  det ⁡ ( S T ) = det ⁡ ( S ) ⋅ det ⁡ ( T ).
• If T is invertible then

the determinant of the inverse is the inverse of the determinant
det ⁡ ( {T}^{− 1} ) = ( \ det ⁡ ( T ) \ {)}^{− 1}.

Matrices T and {T}^{′} are
similar if there is a
nonsingular matrix P such that {T}^{′} = P T {P}^{− 1}.
(This definition is in Chapter Five.)
Show that similar {}2 \ × \ 2 matrices have the same
determinant.

This exercise is recommended for all readers.

Problem 14

Prove that the area of this region in the plane

is equal to the value of this determinant.

det ⁡ ( ( \begin{matrix}{x}_{1} & {x}_{2} \\ {y}_{1} & {y}_{2}\end{matrix} ) )

Compare with this.

det ⁡ ( ( \begin{matrix}{x}_{2} & {x}_{1} \\ {y}_{2} & {y}_{1}\end{matrix} ) )

Problem 15

Prove that for {}2 \ × \ 2 matrices, the determinant of a matrix
equals the determinant of its transpose.
Does that also hold for {}3 \ × \ 3 matrices?

This exercise is recommended for all readers.

Problem 16

Is the determinant function linear — is
det ⁡ ( x ⋅ T + y ⋅ S ) = x ⋅ det ⁡ ( T ) + y ⋅ det ⁡ ( S )?

Problem 17

Show that if A is {}3 \ × \ 3 then
det ⁡ ( c ⋅ A ) = {c}^{3} ⋅ det ⁡ ( A ) for any scalar c.

Problem 18

Which real numbers θ make

( \begin{matrix}cos ⁡ θ & − sin ⁡ θ \\ sin ⁡ θ & cos ⁡ θ\end{matrix} )

singular?
Explain geometrically.

? Problem 19

If a third order determinant has elements
1, 2, ..., 9, what is the maximum value it may
have? (Haggett & Saunders 1955)

Solutions

References

• Haggett, Vern (proposer); Saunders, F. W. (solver) (1955), "Elementary problem 1135", American Mathematical Monthly, 62 (5), American Mathematical Society: 257 {{citation}}: Unknown parameter |month= ignored (help).

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