DLA Discovery guide

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Discovery guide 10.1 Discovery guide

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Reminder.

The effects of the elementary row operations on the determinant are:

swapping rows: $\det(\text{new}) = -\det(\text{old})\text{;}$
multiplying a row by constant $k$: $\det(\text{new}) = k\det(\text{old})\text{;}$
adding a multiple of one row to another: $\det(\text{new}) = \det(\text{old})\text{.}$

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Discovery 10.1.

Consider the general $3\times 3$ matrix

$\begin{equation*} A = \begin{bmatrix} a_{11} \amp a_{12} \amp a_{13} \\ a_{21} \amp a_{22} \amp a_{23} \\ a_{31} \amp a_{32} \amp a_{33} \end{bmatrix}\text{.} \end{equation*}$

Each entry $a_{ij}$ has a corresponding cofactor $C_{ij}\text{,}$ creating a matrix of cofactors

$\begin{equation*} C_A = \begin{bmatrix} C_{11} \amp C_{12} \amp C_{13} \\ C_{21} \amp C_{22} \amp C_{23} \\ C_{31} \amp C_{32} \amp C_{33} \end{bmatrix}\text{.} \end{equation*}$

The transpose of this matrix is called the (classical) adjoint of $A\text{.}$

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(a)

Write out the $(1,1)$ entry of the product $A\utrans{C}_A$ as a formula in the entries of $A$ and $C_A\text{.}$ Does the result look familiar?

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(b)

What do you think the other diagonal entries of $A\utrans{C}_A$ are?

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(c)

Write out the $(1,2)$ entry of the product $A\utrans{C}_A$ as a formula in the entries of $A$ and $C_A\text{.}$ Does the result look familiar? What did we discover about “mixed” cofactor expansions in Discovery 9.6 and Subsection 9.2.3?

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(d)

What do you think the other non-diagonal entries of $A\utrans{C}_A$ are?

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(e)

Refer back to Discovery 8.1. Have we finally achieved Goal 8.3.1?

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Discovery 10.2.

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(a)

Suppose $\det A = 0\text{.}$ If you apply some elementary row operation to $A\text{,}$ what is the determinant of the new matrix? (Consider each of the three kinds of operations.)

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(b)

If $\det A = 0$ and you perform a whole sequence of row operations to $A\text{,}$ what is the determinant of the last matrix in the sequence?

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(c)

Recall that if $A$ is invertible, then it can be row reduced to $I$ (Theorem 6.5.2). If $\det A = 0\text{,}$ could $A$ be invertible?

Hint

Use your answer to Task b.

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(d)

Conversely, if $A$ is invertible, could $\det A = 0\text{?}$

Hint

No need to think about row reducing — combine your answer to Task c with some logical thinking.

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Discovery 10.3.

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(a)

Suppose $\det A \neq 0\text{.}$ Is there any elementary row operation you can apply to $A$ so that the new matrix has determinant $0\text{?}$ (Consider each of the three kinds of operations.)

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(b)

If $\det A \neq 0$ and you perform a whole sequence of row operations to $A\text{,}$ could the last matrix in the sequence have determinant $0\text{?}$

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(c)

Recall that if a matrix is singular (that is, not invertible), then it is not possible to row reduce it to $I$ (Theorem 6.5.2), and so its RREF must have a row of zeros. If $\det A \neq 0\text{,}$ could $A$ be singular?

Hint

Use your answer to Task b.

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(d)

Conversely, if $A$ is singular, is $\det A \neq 0$ possible?

Hint

No need to think about row reducing — combine your answer to Task c with some logical thinking.

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Discovery 10.4.

Recall that for matrix $A$ and elementary matrix $E\text{,}$ the result of $E A$ is the same as the result of performing an elementary row operation on $A$ (namely, the operation corresponding to $E$ ). Verify the formula

$\begin{gather} \det (E A) = (\det E) (\det A) \label{equation-more-det-elem-multiplicative}\tag{$\star$} \end{gather}$

for each of the three types of elementary matrices $E$ (assuming $A$ to be a square matrix of the same size as $E$ ).

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Helpful notes.

To verify a formula, consider LHS and RHS separately, and argue that they equal the same value. Do not work with the proposed equality directly, since you don't know it's an equality yet.
Do not just use examples; think abstractly instead.
For each type of $E\text{,}$ on the LHS consider the product of matrices $E A$ and how its determinant compares to $\det A$ using the rules for how row operations affect determinant (explored in Discovery guide 9.1, and recalled for you at the top of this activity section). For this, think of $\det A = \det (\text{old})$ and $\det (E A) = \det (\text{new})\text{.}$ Then, on the RHS, consider the value of $\det E$ and the corresponding product of numbers $(\det E) (\det A)\text{.}$

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Discovery 10.5.

In this activity, we will verify the general formula

$\begin{gather} \det (M N) = (\det M) (\det N) \label{equation-more-det-multiplicative}\tag{$\star\star$} \end{gather}$

in the case that $M$ is invertible (assuming $M$ and $N$ to be square matrices of the same size).

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(a)

Recall that every invertible matrix can be expressed as a product of elementary matrices (Theorem 6.5.2). For now, suppose that $M$ (which we have assumed invertible) can be expressed as a product of three elementary matrices, say $M = E_1 E_2 E_3\text{.}$ Use formula ( $\star$ ) to verify that

$\begin{equation*} \det (E_1 E_2 E_3 N) = (\det E_1) (\det E_2) (\det E_3) (\det N) \text{.} \end{equation*}$

Hint

Start with the LHS and apply formula ($\star$) one step at a time. In applying formula ($\star$), what are you using for $E$ and for $A$ at each step?

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(b)

Now use formula ( $\star$ ) to verify that

$\begin{equation*} (\det E_1) (\det E_2) (\det E_3) (\det N) = \bbrac{\det (E_1 E_2 E_3)} (\det N). \end{equation*}$

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(c)

Make sure you understand why parts (a) and (b) together verify formula ( $\star\star$ ) for this $M\text{.}$

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(d)

Do you think the calculations in this activity would work out similarly no matter how many $E_i$ 's are required to express $M$ as a product of elementary matrices?

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Discovery 10.6.

If matrix $A$ is invertible, by definition this means that $A \inv{A} = I$ (as well as $\inv{A} A = I$ ).

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(a)

Determine the value of $\det (A \inv{A})$ from the equality $A \inv{A} = I\text{.}$

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(b)

Starting with your answer to Task a, use formula ( $\star\star$ ) from Discovery 10.5 to obtain a formula for $\det (\inv{A})$ in terms of $\det A\text{.}$

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Discovery 10.7.

In this discovery activity, we extend formula ( $\star\star$ ) to also be valid in case that $M$ is singular (assuming $M$ and $N$ to be square matrices of the same size).

Recall that if $M$ is singular (i.e. not invertible), then every product $M N$ is singular (Statement 1 of Proposition 6.5.8).

Combine this with your answer to Discovery 10.3.d to verify formula ( $\star\star$ ) in the case that $M$ is singular.