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Questions for Math-1: A course for Biology and Applied Geology students

Abhay Soman

Exercises 1.2 Exercises

A few exercises related to basic matrix operations are listed below.

1.

Consider the following matrices in \(M_{3\times 2}(\mathbb{R})\text{.}\)
\begin{equation*} A=\begin{pmatrix}1\amp 9\\\tfrac{-1}{2}\amp -1\\2\amp -2\end{pmatrix},\quad B=\begin{pmatrix}1\amp 9\\\tfrac{-1}{2}\amp -1\\\tfrac{1}{9}\amp \tfrac{-1}{2}\end{pmatrix},\quad\text{and}\quad C=\begin{pmatrix}0\amp -5\\\tfrac{-1}{10}\amp 10\\2\amp -2\end{pmatrix} \end{equation*}
Find \(2A+2B-5C\text{.}\)

2.

Let \(A=\left(\begin{smallmatrix}1\amp 3\\0\amp 4\end{smallmatrix}\right)\in M_{2}(\mathbb{R})\text{.}\) Find \(B\in M_{2}(\mathbb{R})\) such that \(A+100B=\left(\begin{smallmatrix}1\amp 0\\1\amp 0\end{smallmatrix}\right)\text{.}\)

3.

Show that for any matrix \(A\in M_{n}(\mathbb{R})\) the following is true.
\begin{equation*} A+0_n=0_n+A=A\quad\text{and}\quad A\, I_n=I_n\, A=A,\quad\text{and}\quad A\,0_n=0_n\,A=0_n \end{equation*}

4.

Show that for any matrices \(A,B,C\in M_{5\times 7}(\mathbb{R})\) the following is true.
\begin{equation*} (A+B)+C=A+(B+C) \end{equation*}

5.

Let \(A,B,C\in M_4(\mathbb{R})\text{.}\) Show that \(A(B+C)=AB+AC\) and that \((A+B)C=AC+BC\text{.}\)

6.

Compute the matrix multiplication \(AB\text{.}\)
  1. \(A=\begin{pmatrix}1\amp 2\amp -3\\-1\amp 0\amp 4\end{pmatrix}\) and \(B=\begin{pmatrix}1\amp 2\amp 3\amp 0\\0\amp 0\amp 1\amp -1\\-1\amp 0\amp -2\amp -1\end{pmatrix}\)
  2. \(A=\begin{pmatrix}-2\amp -4\amp 0\amp -2\\-5\amp -2\amp -11\amp 7\end{pmatrix}\) and \(B=\begin{pmatrix}1\\-1\\1\\-1\end{pmatrix}\)
  3. \(\displaystyle A=B=\begin{pmatrix}1\amp -1\\0\amp 0\end{pmatrix}\)
  4. \(A=\begin{pmatrix}1\amp 2\amp -2\end{pmatrix}\) and \(B=\begin{pmatrix}-2\\1\\0\end{pmatrix}\)
    In this case, consider the matrix \(A\) as the point \(P_A=(1,2,-2)\) of the space and matrix \(B\) as the point \(P_B=(-2,1,0)\text{.}\) Plot the line segment \(OP_A\) joining the origin of \(\mathbb{R}^3\) and \(P_A\) as well as the line segment \(OP_B\) joining the origin of \(\mathbb{R}^3\) and \(P_B\text{.}\) Check whether \(OP_A\) is perpendicular to \(OP_B\text{.}\)
  5. \(A=\begin{pmatrix}-1\amp -1\amp -1\end{pmatrix}\) and \(B=\begin{pmatrix}-1\\0\\0\end{pmatrix}\)
    In this case, consider the matrix \(A\) as the point \(P_A=(-1,-1,-1)\) of the space and matrix \(B\) as the point \(P_B=(-1,0,0)\text{.}\) Plot the line segment \(OP_A\) joining the origin of \(\mathbb{R}^3\) and \(P_A\) as well as the line segment \(OP_B\) joining the origin of \(\mathbb{R}^3\) and \(P_B\text{.}\) Check whether \(OP_A\) is perpendicular to \(OP_B\text{.}\)

7.

For a matrix \(A\in M_n(\mathbb{R})\) and a natural number \(k\) we write
\begin{equation*} A^k=\underbrace{AAA\cdots A}_{k\text{-times}}. \end{equation*}
In particular, \(A^1=A\) and \(A^2=A\,A\text{.}\)
Find the following powers of the given matrices.
\begin{equation*} \begin{pmatrix}1\amp 6\\0\amp 1\end{pmatrix}^3;\quad\begin{pmatrix}0\amp -1\\1\amp 0\end{pmatrix}^4;\quad\begin{pmatrix}1\amp -1\amp 0\\0\amp 1\amp -1\\0\amp 0\amp 1\end{pmatrix}^2;\quad\begin{pmatrix}0\amp 1\amp 1\\0\amp 0\amp -1\\0\amp 0\amp 0\end{pmatrix}^3 \end{equation*}

8.

Write the augmented matrix corresponding to the given system of equations.
  1. \begin{align*} 2x+y-z \amp =2\\ x+y+z \amp =0 \end{align*}
  2. \begin{align*} -x+3y+z \amp =-2\\ x+y+9z \amp =0\\ 2x+8y \amp =30 \end{align*}

9.

In the following, given \(A\) compute its transpose and also compute \(A\,A^t\) and \(A^t\, A\text{.}\)
  1. \(A=\begin{pmatrix}a\amp b\amp c\end{pmatrix}\in M_{1\times 3}(\mathbb{R})\text{,}\) where \(a,b,c\in\mathbb{R}\)
  2. \(A=\begin{pmatrix}a\\b\\c\\d\\e\end{pmatrix}\in M_{5\times 1}(\mathbb{R})\text{,}\) where \(a,b,c,d,e\in\mathbb{R}\)
  3. \(\displaystyle A=\begin{pmatrix}1\amp 2\amp 3\\-1\amp -2\amp -3\\0\amp 0\amp 0\end{pmatrix}\in M_3(\mathbb{R})\)
  4. \(\displaystyle A=\begin{pmatrix}0\amp 1\amp 0\\1\amp 0\amp 0\\0\amp 0\amp 1\end{pmatrix}\in M_3(\mathbb{R})\)
  5. \(\displaystyle A=\begin{pmatrix}-1\amp 0\amp 0\amp 0\\0\amp 0\amp 0\amp -1\\0\amp -1\amp 0\amp 0\\0\amp 0\amp -1\amp 0\end{pmatrix}\in M_4(\mathbb{R})\)
  6. Find a condition on \(a,b,c,d\in\mathbb{R}\) such that the matrix \(A=\begin{pmatrix}a\amp b\\c\amp d\end{pmatrix}\) will be equal to its transpose.
  7. Consider a matrix \(A=\begin{pmatrix}a\amp b\\c\amp d\end{pmatrix}\text{.}\) Find a condition on \(a,b,c,d\in\mathbb{R}\) such that \(A\,A^t=I_2\) and \(A^t\,A=I_2\text{.}\)

10.

Check whether \(f\) is a function or not. Also check whether \(f\) is injective (one-one) or surjective (onto).
\begin{equation*} f\colon M_2(\mathbb{R})\to\mathbb{R}\quad\text{given by}\quad\begin{pmatrix}a\amp b\\c\amp d\end{pmatrix}\mapsto ad-bc. \end{equation*}

11.

Find the inverse of the given matrix.
  1. \(\displaystyle \begin{pmatrix}1\amp 0\\0\amp 1\end{pmatrix}\)
  2. \(\displaystyle \begin{pmatrix}1\amp 1\\0\amp 1\end{pmatrix}\)
  3. \(\displaystyle \begin{pmatrix}0\amp 1\\-1\amp 0\end{pmatrix}\)
  4. \(\begin{pmatrix}a\amp 0\\0\amp b\end{pmatrix}\text{,}\) where \(a\) and \(b\) are nonzero real numbers.

12.

In each of the following system \(AX=B\text{,}\) the matrices \(A,B\) are given. Find the matrix \(X\text{.}\)
  1. \begin{equation*} A=\begin{pmatrix}-1\amp 0\\-1\amp 1\end{pmatrix}\quad \text{and}\quad B=\begin{pmatrix}0\\1\end{pmatrix}\text{.} \end{equation*}
    Find \(X=\begin{pmatrix}x_1\\x_2\end{pmatrix}\text{.}\)
  2. \begin{equation*} A=\begin{pmatrix}5\amp 2\\-2\amp 0\end{pmatrix}\quad \text{and}\quad B=\begin{pmatrix}7\\5\end{pmatrix}\text{.} \end{equation*}
    Find \(X=\begin{pmatrix}x_1\\x_2\end{pmatrix}\text{.}\)

13.

Show that if the inverse of a matrix \(A\in M_n(\mathbb{R})\) exists then it is unique, i.e., if \(B\in M_n(\mathbb{R})\) and \(C\in M_n(\mathbb{R})\) are such that \(AB=I_n=BA\) and \(AC=I_n=CA\text{,}\) then \(B=C\text{.}\)
Hint.
Multiply each side of the equation \(AB=I_n\) on the left by \(C\text{.}\)

14.

    Multiplication \(AB\) is defined when \(A\in M_{3\times 55}(\mathbb{R})\) and \(B\in M_{55\times 1}(\mathbb{R})\text{.}\)

  • True.

  • False.

15.

    For any two matrices \(A,B\in M_{6}(\mathbb{R})\text{,}\) \(AB=BA\text{.}\)

  • True.

  • False.

16.

    For any \(a\in\mathbb{R}\text{,}\) \(\left(\begin{smallmatrix}0\amp a\\0\amp 0\end{smallmatrix}\right)^2=\left(\begin{smallmatrix}0\amp 0\\0\amp 0\end{smallmatrix}\right)\in M_2(\mathbb{R})\text{.}\)

  • True.

  • False.

17.

    The following system of linear equations has infinitely many solutions.
    \begin{align*} x+y\amp = 2\\ z\amp =0 \end{align*}

  • True.

  • False.

18.

    For the following matrix \(A\in M_3(\mathbb{R})\) we have \(A=-A^t\text{.}\)
    \begin{equation*} A=\begin{pmatrix}0\amp 1\amp 1\\-1\amp 0\amp 0\\-1\amp -1\amp 1\end{pmatrix}\in M_3(\mathbb{R}) \end{equation*}

  • True.

  • False.

19.

    Every matrix in \(M_3(\mathbb{R})\) has an inverse.

  • True.

  • False.

20.

    Let \(A,B\in M_n(\mathbb{R})\text{.}\) The inverse of \(AB\) is \(B^{-1}A^{-1}\text{,}\) i.e.,
    \begin{equation*} \left(AB\right)^{-1}=B^{-1}A^{-1}. \end{equation*}

  • True.

  • False.

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