Discovery 9.1.
What is \(\det A\) if \(A\) is a square matrix with a row of zeros? Explain by referring to a cofactor expansion.
Discovery guide 9.1 Discovery guide
Discovery 9.2.
Consider the matrix
\begin{equation*}
\begin{bmatrix} 1 \amp 1 \amp 1 \\ 2 \amp 1 \amp 1 \\ 3 \amp 1 \amp 0 \end{bmatrix}.
\end{equation*}
(a)
Compute the determinant by cofactor expansion along the first row.
(b)
Now swap the first and second rows, and compute the determinant of the new matrix by cofactor expansion along the second row (which will now have the entries of first row of the original matrix). Why do you think you got the answer you did?
Hint.
Do you remember the cofactor sign patterns? If not, see Pattern (8.3.1).
(c)
Do you think the same thing will happen if you swap the second and third rows of the original matrix?
(d)
What about if you swap the first and third rows of the original matrix?
(e)
What if you swap the \(\irst\) and \(\ond\) rows of the original matrix, then swap the \(\ond\) and \(\ird\) rows of that matrix, and then swap the \(\irst\) and \(\ond\) rows of that matrix? Do you want to change your answer to Task d?
(f)
Complete the rule: If \(B\) is obtained from \(A\) by swapping two rows, then \(\det B\) is related to \(\det A\) by .
(g)
Complete the rule: If \(E\) is an elementary matrix of the “swap two rows” type, then \(\det E = \fillinmath{XXXXXX}\text{.}\)
Hint.
How do you create an elementary matrix?
Discovery 9.3.
Suppose \(A\) is a square matrix with two identical rows. What happens to the matrix when you swap those two identical rows? According to Discovery 9.2, what is supposed to happen to the determinant when you swap rows? What can you conclude about \(\det A\text{?}\)
Discovery 9.4.
Consider the matrix from Discovery 9.2.
(a)
Multiply the first row by \(7\text{,}\) and compute the determinant of the new matrix. Do you think the same will happen if you multiplied some other row of the matrix by \(7\text{?}\) Explain by referring to cofactor expansions.
(b)
Complete the rule: If \(B\) is obtained from \(A\) by multiplying one row by \(k\text{,}\) then \(\det B\) is related to \(\det A\) by .
(c)
Complete the rule: If \(E\) is an elementary matrix of the “multiply a row by \(k\)” type, then \(\det E =\fillinmath{XXXXXX}\text{.}\)
Hint.
How do you create an elementary matrix?
(d)
Suppose you multiply the whole matrix by \(7\text{.}\) What happens to the determinant in that case?
Hint.
How many rows are you multiplying by \(7\text{?}\)
(e)
Complete the rule: For scalar \(k\) and \(n\times n\) matrix \(A\text{,}\) \(\det (kA) = \fillinmath{XXXXX}\text{.}\)
Hint.
If you multiply a whole matrix by a scalar, you are in effect multiplying every row by that scalar.
Discovery 9.5.
Suppose \(A\) is a square matrix where one row is equal to a multiple of another. Combine your answer to Discovery 9.3 with a rule from Discovery 9.4 to determine \(\det A\text{.}\)
Discovery 9.6.
Consider the generic \(3\times 3\) matrix
\begin{equation*}
\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}.
\end{equation*}
Its determinant is \(a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\text{.}\)
Suppose we add \(k\) times the second row to the first:
\begin{equation*}
\begin{bmatrix}
a_{11}+k a_{21} \amp a_{12}+k a_{22} \amp a_{13}+k a_{23} \\
a_{21} \amp a_{22} \amp a_{23} \\
a_{31} \amp a_{32} \amp a_{33}
\end{bmatrix}.
\end{equation*}
(a)
Has this row operation changed the cofactors of entries in the first row?
(b)
Write out the cofactor expansion along the first row for the new matrix. Then use some algebra to express this cofactor expansion as:
\begin{equation*}
(\text{some formula}) + k (\text{some other formula}).
\end{equation*}
The first “some formula” should look familiar. Can you craft a \(3 \times 3\) matrix so that “some other formula” can be similarly interpreted?
(c)
What is the value of the “some other formula” part from Task b?
Hint.
(d)
Complete the rule: If \(B\) is obtained from \(A\) by adding a multiple of one row to another, then \(\det B\) is related to \(\det A\) by .
(e)
Complete the rule: If \(E\) is an elementary matrix of the “add a multiple of one row to another” type, then \(\det E = \fillinmath{XXXXXX}\text{.}\)
Hint.
How do you create an elementary matrix?
