Chapter 2 – Quiz 1 – E600 Mathematics

1. Which statements about matrix characteristics are true? (Multiple choice)

If a matrix is upper triangular and symmetric, then it is diagonal.

A zero matrix can not be a square matrix.

Any diagonal matrix has only non-zero entries on the diagonal.

If $A$ is a square matrix, then $A'=A$ .

There are diagonal matrices with strictly more rows than columns.

Every identity matrix is lower triangular.

2. Which property of “regular” (i.e., real number) multiplication does generally not transfer to matrix multiplication ( $a, b, c\in\mathbb R$ )? (Single choice)

Distributivity over addition: $a\cdot (b + c) = a\cdot b + a \cdot c$ .

Symmetry: $a \cdot b = b\cdot a$ .

Multiplication by zero: $a \cdot 0 = 0$ .

Associativity: $a\cdot (b\cdot c) = (a\cdot b) \cdot c$ .

3. Which statements about matrix characteristics and calculus are true? (Multiple choice)

If $A\in\mathbb R^{n\times m}, B\in\mathbb R^{m\times k}$ and $k\neq n$ , then $B\cdot A$ exists.

If $v$ is a column vector of length $n>1$ , then $v'v$ is a matrix of dimension $n\times n$ .

If $A\in\mathbb R^{n\times m}, B\in\mathbb R^{m\times k}$ , then $A\cdot B$ exists.

If $A,B$ are matrices for which $A\cdot B$ exists, and $B$ is a zero matrix, then $A\cdot B$ is also a zero matrix.

If $A$ is a symmetric matrix, then $A'=A$ .

If $v$ is a column vector of length $n>1$ , then it is always true that $v'v\geq 0$ .

4. What do you obtain from multiplying out $v'A'Bv$ when $v = \begin{pmatrix} 2\\-1\end{pmatrix}$ , $A = \begin{pmatrix} 4 &1 \\0 & 2\\-1 & 0\end{pmatrix}$ and $B = \begin{pmatrix} 1 &2 \\-3 & 4\\5 & -6\end{pmatrix}$ ? (Write down the result as a number, do not use letters, spaces, punctuation, etc.)

5. If $A\in\mathbb R^{4\times 3}$ and $B\in\mathbb R^{4\times 9}$ , what is the dimension of $B'A$ ?

The product does not exist.

$4\times 9$ .

$9\times 3$ .

$4\times 3$ .

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