Question

Which of the following statements are true about scalar multiplication of matrices?

Answer 1

Consider all options:

1. You can multiply a matrix of any size by a scalar. This option is true, because the rule of multiplication matrix by the number k, where k≠0 (according to definition) is

$k\cdot \left(\begin{array}{ccc}a_{11} & \dots & a_{1n}\\& \dots & \\a_{m1} & \dots & a_{mn}\end{array}\right)=\left(\begin{array}{ccc}ka_{11} & \dots & ka_{1n}\\& \dots & \\ka_{m1} & \dots & ka_{mn}\end{array}\right).$

2. For any matrix A, 1 × A = A. This option is true, because according to previous definition

$1\cdot A=1\cdot \left(\begin{array}{ccc}a_{11} & \dots & a_{1n}\\& \dots & \\a_{m1} & \dots & a_{mn}\end{array}\right)=\left(\begin{array}{ccc}1\cdot a_{11} & \dots & 1\cdot a_{1n}\\& \dots & \\1\cdot a_{m1} & \dots & 1\cdot a_{mn}\end{array}\right)=\left(\begin{array}{ccc}a_{11} & \dots & a_{1n}\\& \dots & \\a_{m1} & \dots & a_{mn}\end{array}\right)=A.$

3. For any scalar r, rI = I, where I is the identity matrix. This option is false, for example

$r\cdot \left(\begin{array}{cc}1 &0\\0 & 1\end{array}\right)= \left(\begin{array}{cc}r &0\\0 & r\end{array}\right)\neq \left(\begin{array}{cc}1 &0\\0 & 1\end{array}\right).$

4. You can scale geometric figures using scalar multiplication. Geometric figures consists of points and their coordinates can be considered as matrices. Using scalar multiplication to these matrices you'll get coordinates of image figures.

This option is true.

5. Scalar multiplication is a shortcut for repeated addition of the same matrix. While adding matrices you add the corresponding numbers ($a_{ij}$ to $a_{ij}$), then for each matrix's element you can use  multiplication as a shortcut for repeated addition and then get  multiplication as a shortcut for repeated addition for whole matrix.

6. Scalar multiplication is not possible for matrices that are not square. This is not true, because the definition says that multiplication matrix by a scalar is possible for all matrices.

Answer: correct options are 1, 2, 4 and 5.

Answer 2

Scalar multiplication of a matrix refers to the multiplication of all the elements of the matrix by an ordinary number which is called a scalar.

From the given options the statements which are true about scalar multiplication of matrices are;

a)       You can multiply a matrix of any size by a scalar.

b)      For any matrix A, 1 × A = A.

c)       Scalar multiplication is a shortcut for repeated addition of the same matrix.