Chapter 2 – Quiz 1 – E600 Mathematics

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

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

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

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

2. 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.)

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

$4\times 3$ .

$9\times 3$ .

The product does not exist.

$4\times 9$ .

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

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

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

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

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

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

Every identity matrix is lower triangular.

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

A zero matrix can not be a square matrix.

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

There are diagonal matrices with strictly more rows than columns.

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

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