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Let \(A = \left [ \begin{array}{rrr} 1 & 2 & 4 \\ 0 & 1 & 3 \\ -2 & 5 & 1 \end{array} \right ]\). When doing cofactor expansion along the top row, we encounter three minor matrices. What are they?
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of
the determinant.
\begin{equation*} \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \rightarrow \cdots \rightarrow \left [ \begin{array}{cc} a & c \\ b & d \end{array} \right ] \end{equation*}
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It does not change the determinant. This was just taking the transpose.
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of
the determinant.
\begin{equation*} \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \rightarrow \cdots \rightarrow \left [ \begin{array}{cc} c & d \\ a & b \end{array} \right ] \end{equation*}
Click the arrow to see answer.
In this case two rows were switched and so the resulting determinant is \(-1\) times the first.
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of
the determinant.
\begin{equation*} \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \rightarrow \cdots \rightarrow \left [ \begin{array}{cc} a & b \\ a+c & b+d \end{array} \right ] \end{equation*}
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The determinant is unchanged. It was just the first row added to the second.
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of
the determinant.
\begin{equation*} \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \rightarrow \cdots \rightarrow \left [ \begin{array}{cc} a & b \\ 2c & 2d \end{array} \right ] \end{equation*}
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The second row was multiplied by 2 so the determinant of the result is 2 times the original
determinant.
An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of
the determinant.
\begin{equation*} \left [ \begin{array}{cc} a & b \\ c & d \end{array} \right ] \rightarrow \cdots \rightarrow \left [ \begin{array}{cc} b & a \\ d & c \end{array} \right ] \end{equation*}
Click the arrow to see answer.
In this case the two columns were switched so the determinant of the second is \(-1\) times the
determinant of the first.
Let \(A\) be an \(n\times n\) matrix and suppose there are \(n-1\) rows such that all rows are linear combinations of these \(n-1\) rows. Show
\(\det \left ( A\right ) =0\).
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If the determinant is nonzero, then it will remain nonzero with row operations applied to the
matrix. In this case, you can obtain a row of zeros by doing row operations. Thus the determinant must be zero.
Construct \(2\times 2\) matrices \(A\) and \(B\) to illustrate the property that \(\det A \det B = \det (AB)\).
An \(n\times n\) matrix is called nilpotent if for some positive integer, \(k\) it follows \(A^{k}=O.\) If \(A\) is a nilpotent matrix and \(k\) is the smallest possible
integer such that \(A^{k}=O,\) what are the possible values of \(\det \left ( A\right )\)?
A matrix is said to be orthogonal if \(A^{T}A=I.\) Thus the inverse of an orthogonal matrix is just its transpose. What are the possible
values of \(\det \left ( A\right ) \) if \(A\) is an orthogonal matrix?
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You would need \(\det \left ( AA^{T}\right ) =\det \left ( A\right ) \det \left ( A^{T}\right ) =\det \left ( A\right ) ^{2}=1\) and so \(\det \left ( A\right ) =1,\) or \(-1\).
Let \(A\) and \(B\) be two \(n\times n\) matrices. We say that \(A\) is similar to \(B\) and write \(A\sim B\) provided that there exists an invertible matrix \(P\) such that \(A=P^{-1}BP.\)
Show that if \(A\sim B,\) then \(\det \left ( A\right ) =\det \left ( B\right )\).
Determine whether each statement is true or false. If true, provide a proof. If false, provide a counter example.
1.
If any two columns of a square matrix are equal, then the determinant of the matrix equals zero.
2.
If \(A^{-1}\) exists then \(\det \left ( A^{-1}\right ) =\det \left ( A\right ) ^{-1}.\)
3.
If \(A\) is a real \(n\times n\) matrix, then \(\det \left ( A^{T}A\right ) \geq 0.\)
4.
If \(AX=0\) for some \(X \neq 0,\) then \(\det \left ( A\right ) =0.\)
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All of the statements are true.
1.
\(\det (A)=\det (A^T)\). If two columns of \(A\) are equal, then two rows of \(A^T\) are equal. Applying row operations to \(A^T\) will produce a matrix
with a row of zeros.
Divide both right and left side by \(\det (A)\) to obtain the result.
3.
This follows from the fact that \(\det (A)=\det (A^T)\).
4.
If \(AX=0\) for some \(X \neq \vec {0}\) then there is a non-trivial linear combination of the columns of \(A\) that is equal to \(\vec {0}\). This means that
the columns of \(A\) are linearly dependent. This implies that the rows are also linearly dependent (why?). Applying
elementary row operations will lead us to a row of zeros.
Additional Problems