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\begin{equation*} A = \left [ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos t & -\sin t \\ 0 & \sin t & \cos t \end{array} \right ] \end{equation*}
Does there exist a value of \(t\) for which this matrix fails to have an inverse? Explain.
Click the arrow to see answer.
No. It has a nonzero determinant for all \(t\).
If \(A,B,\) and \(C\) are each \(n\times n\) matrices and \(ABC\) is invertible, show why each of \(A,B,\) and \(C\) are invertible.
Click the arrow to see answer.
This follows because \(\det \left ( ABC\right ) =\det \left ( A\right ) \det \left ( B\right ) \det \left ( C\right ) \) and if this product is nonzero, then each determinant in the product is nonzero and so each of these
matrices is invertible.
Suppose \(A,B\) are \(n\times n\) matrices and that \(AB=I.\) Show that then \(BA=I.\)
First explain why \(\det \left ( A\right ) ,\det \left ( B\right ) \) are both nonzero. Then \(\left ( AB\right ) A=A\) and then show \(BA\left ( BA-I\right ) =0.\) Now use
what is given to conclude \(A\left ( BA-I\right ) =0.\)
Suppose \(A\) is an upper triangular matrix. Show that \(A^{-1}\) exists if and only if all elements of the main diagonal are non zero. Is it
true that \(A^{-1}\) will also be upper triangular? Explain. Could the same be concluded for lower triangular matrices?
The given
condition is what it takes for the determinant to be non zero. Recall that the determinant of an upper triangular matrix is just
the product of the entries on the main diagonal.
Let \(A\), \(B\), and \(C\) denote \(n\times n\) matrices. Assume that \(\det A=-1\), \(\det B=2\), and \(\det C=3\). Evaluate
1.
\(\det (A^3BC^TB^{-1})\)
2.
\(\det (B^2C^{-1}AB^{-1}C^T)\)
3.
\(\det (A^{-1}B^{-1}AB)\)
If \(A\) and \(B\) are \(n\times n\) matrices such that \(AB =-BA\), and if \(n\) is odd, show that either \(A\) or \(B\) has no inverse.
Show that no \(3\times 3\) matrix \(A\) exists such that \(A^2+I = O\). Find a \(2\times 2\) matrix \(A\) with this property.
Show that \(\det (A+B^T ) = \det (A^T +B)\) for any \(n\times n\) matrices \(A\) and \(B\).
