Answer each of the following questions.
If \(\ABld\) is an \(m \times n\) matrix, how many columns does it have? How many rows?
When can you not add two matrices together?
When can you not multiply two matrices together?
Is it always the case that \({\bf AB} = {\bf BA}\)?
What is \({\bf I}\)? Given an \(m \times n\) matrix \({\bf A}\) and an \(n \times n\) matrix \({\bf I}\), what is \({\bf AI}\) and what are its dimensions? Does \({\bf AI} = {\bf IA}\)?
For problems 2-4, let\[{\bf A} = \begin{pmatrix} 1 & 2 & 1 \\ 3 & 1 & 1 \\ 2 & 1 & 2 \end{pmatrix} \,\hspace{0.7in} {\bf B} = \begin{pmatrix} 4 & 1 & 5 \\ 1 & 2 & 1 \\ 3 & 1 & 4 \end{pmatrix} \,.\]Calculate the following.
\({\bf A} + {\bf B}\)
\(2{\bf A} + {\bf B}\)
\({\bf A} - {\bf B}\)
For problems 5-7, let\[{\bf A} = \begin{pmatrix} 2 & 1 \\ 8 & 4 \end{pmatrix} \hspace{0.7in} {\bf B} = \begin{pmatrix} 3 & 1\\ 2 & 1 \end{pmatrix}\,.\]Calculate the following.
\({\bf AB}\)
\({\bf BA}\)
\(2{\bf A} - {\bf AB}\)
For problems 8-13 let,\[{\bf A} = \begin{pmatrix} 1+i & 2 \\ i & 1-3i \end{pmatrix} \,\hspace{0.7in} {\bf B} = \begin{pmatrix} i & 0 \\ 0 & 2i \end{pmatrix} \,.\]Calculate the following.
\({\bf A}^{T}\)
\({\bf \overline A}\)
\({\bf A}^{*}\)
\(({\bf AB})^{T}\)
\({\bf B}^{T}{\bf A}^{T}\)
\(({\bf AB})^{*}\)
Show the following properties of invertibility as mentioned in the text. If \(\ABld\) and \(\BBld\) are \(n\times n\) invertible matrices and \(\alpha \neq 0\), then
\(\displaystyle(\alpha\ABld)^{-1} = \frac{1}{\alpha}\ABld^{-1}\)
\((\ABld\BBld)^{-1} = \BBld^{-1}\ABld^{-1}\)
\((\ABld^{-1})^{-1} = \ABld\)
\((\overline{\ABld})^{-1} = \overline{\ABld^{-1}}\)
\((\ABld^T)^{-1} = (\ABld^{-1})^T\)
\((\ABld^*)^{-1} = (\ABld^{-1})^*\)
Answer the following questions,
What is the definition of an inverse of a matrix \({\bf A}\)?
If det(\({\bf A}) \ne 0\), is the matrix \({\bf A}\) invertible? If so, find \({\bf A}^{-1}\) if \({\bf A} = \begin{pmatrix} a&b\\c&d \end{pmatrix}\).
Show the following properties of determinants as mentioned in the text. If \(\ABld\) is an \(2\times 2\) matrix then
\(\det \left(\overline{\ABld}\right) = \overline{\det\left(\ABld\right)}\)
\(\det \left(\ABld^T\right) = \det \left(\ABld\right)\)
\(\det \left(\ABld^*\right) = \overline{\det\left(\ABld\right)}\)
For problems 17-21 calculate the determinants of the given matrices. State whether they are invertible or not. If invertible, calculate the inverse.
\(\begin{pmatrix} 1&4\\-2&3 \end{pmatrix}\)
\(\begin{pmatrix} 3&-1\\6&1 \end{pmatrix}\)
\(\begin{pmatrix} 10&1\\9&2 \end{pmatrix}\)
\(\begin{pmatrix} 1&1&2\\2&1&1\\1&2&1 \end{pmatrix}\)
\(\begin{pmatrix} 2&3&1\\-1&2&1\\4&-1&-1 \end{pmatrix}\)
For problem 22-24 solve the system of equations
\(\begin{pmatrix} 2&-1\\-1&1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix}6\\-1 \end{pmatrix} \)
\(\begin{pmatrix} 2&3\\1&-3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix}6\\-15 \end{pmatrix} \)
\(\begin{pmatrix} 1&4\\3&7 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix}7\\6 \end{pmatrix} \)
For problems 25-26 use row reduction to solve the system of equations
\(\begin{pmatrix} 1&0&-1\\3&1&1\\-1&1&2 \end{pmatrix} \begin{pmatrix} x \\ y \\z \end{pmatrix} = \begin{pmatrix}0\\1\\2 \end{pmatrix} \)
\(\begin{pmatrix} 1&2&-1\\2&1&1\\1&-1&2 \end{pmatrix} \begin{pmatrix} x \\ y \\z \end{pmatrix} = \begin{pmatrix}2\\1\\-1 \end{pmatrix} \)
Suppose we associate to every complex number \(a+ib\) a matrix
\[\ZBld(a+ib) = \begin{pmatrix} a & b \\ -b & a \end{pmatrix}.\]
If \(z\) and \(w\) are any two complex numbers, verify the following properties:
\(\ZBld(z) + \ZBld(w) = \ZBld(z + w)\)
\(\ZBld(z)\ZBld(w) = \ZBld(zw) = \ZBld(w)\ZBld(z)\)
\(\ZBld(z)^T = \ZBld(a-ib) = \ZBld(\overline{z})\)
\(\ZBld(z)\ZBld(z)^T = (a^2 + b^2)\IBld = \ZBld(z\overline{z})\)
\(\det\ZBld(z) = a^2 + b^2 = z\overline{z}\)
\(\displaystyle \ZBld(z)^{-1} = \frac{1}{a^2 + b^2}\ZBld(z)^T = \ZBld(z^{-1}), \quad z\neq 0.\)
Recall that the derivative of a matrix valued function \(\ABld(t)\) is defined entrywise, \[\frac{\dee \ABld}{\dt} = \left(\frac{\dee a_{ij}}{\dt}\right).\] If \(\ABld(t)\) and \(\BBld(t)\) are matrix valued functions, show that \[\frac{\dee}{\dt}\left(\ABld\BBld\right) = \ABld\frac{\dee\BBld}{\dt} + \frac{\dee\ABld}{\dt}\BBld.\]
Let \(\ABld\) be an upper triangular matrix, that is a matrix of the form \[\ABld = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ 0 & a_{22} & \cdots & a_{2n} \\ \vdots & & \ddots & \vdots\\ 0 & \cdots & 0 & a_{nn} \end{pmatrix}.\] What is \(\det \left(\ABld\right)\)? Give conditions on the entries of \(\ABld\) for it to be invertible.
A matrix \(\ABld\) is refered to as nilpotent if there is a natural number \(n>0\), such that \[\ABld^n = \underbrace{\ABld\ABld \ldots \ABld}_{n\text{ times}} = {\mathbf 0}.\] Show that a nilpotent matrix cannot be invertible. In particular, show that any \(n\times n\) matrix \(\ABld\) of the form \[\ABld = \begin{pmatrix} 0 & a_{12} & a_{13} & \cdots & a_{1n}\\ 0 & 0 & a_{23} & \cdots & a_{2n}\\ \vdots & & \ddots & \ddots & \vdots\\ 0 & \cdots & \cdots & 0 & a_{n-1,n}\\ 0 & \cdots & \cdots & \cdots & 0 \end{pmatrix}\] is nilpotent with \(\ABld^n ={ \mathbf 0}\) (\(n\) being the size of the matrix). Using the formula derived in the previous problem, compute \(\det{\left(\ABld\right)}\) to verify the non-invertibility claim above.