Resolved: "Consider the 3×3 matrices with real entrices. Show..."

College
Algebra
Linear algebra
Matrix transformations

necessaryh

Answered question

2021-09-20

Consider the

3 \times 3

matrices with real entrices. Show that the matrix forms a vector space over R with respect to matrix addition and matrix multiplication by scalars?

Answer & Explanation

stuth1

Skilled2021-09-21Added 97 answers

Step 1
Let $M_{3} (R)$ be the set of all $3 \times 3$ matrices of real entries. Then we have to show that the set $M_{3} (R)$ of all $3 \times 3$ matrices form a vector space with respect to the usual matrix addition and usual scalar multiplication.
We will show that all 10 properties are satisfied for $M_{3} (R)$ to be a vector space as follows;
Suppose $A, B, C \in M_{3} (R),$ where
$A = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]$
$B = [\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix}]$
$C = [\begin{matrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{matrix}]$
and $a_{i j}, b_{i j}, c_{i j} \in R$ for all $1 \leq i, j \leq 3$
(i) Closure:
$A + B = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}] + [\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix}] = [\begin{matrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \\ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \\ a_{31} + b_{31} & a_{32} + b_{32} & a_{33} + b_{33} \end{matrix}]$
Now we know that $a_{i j} + b_{i j} \in R$ for all $1 \leq i < j \leq 3$
So (a) $A + B$ is also $\in M_{3} (R)$
(ii) Associativity:

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