The trace of a square n x n matrix A = (a_ij) is the sum a11 + a22 +...+ ann of the entries on its main diagonal.Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 matrices with real entries that have trace 0. Is H a subspace of the vector space V?Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated list and syntax such as [[????,????],[????,????]], [[????,????],[????,????]] for the answer [1324],[5768]. (Hint: to show that H is not closed under addition, it is sufficient to find two idempotent matrices ???? and ???? such that (????+????)2≠(????+????).)

Answer:Subspace; ClosedStep-by-step explanation:The trace of a square n x n matrix [tex]A = (a_{ij})[/tex] is the sum [tex]a_{11} + a_{22} +...+ a_{nn}[/tex] of the entries on its main diagonal.Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 matrices with real entries that have trace 0.Theorem: H is a subspace of the vector space V, if 1) for every [tex]A,\ B\in H: \ \ A+B\in H;[/tex]2) for each [tex]A\in H[/tex] and [tex]\lambda \in R:\ \ \lambda A\in H.[/tex]Check these two conditions:1) Let [tex]A=(a_{ij}),\ B=(b_{ij})\in H[/tex] This means[tex]a_{11}+a_{22}=0\\ \\b_{11}+b_{22}=0[/tex]Consider the matrix [tex]A+B=(a_{ij}+b_{ij})=\left(\begin{array}{cc}a_{11}+b_{11}&a_{12}+b_{12}\\a_{21}+b_{21}&a_{22}+b_{22}\end{array}\right)[/tex]This matrix sum has the trace[tex](a_{11}+b_{11})+(a_{22}+b_{22})=(a_{11}+a_{22})+(b_{11}+b_{22})=0+0=0[/tex]So, [tex]A+B\in H[/tex]2) Consider [tex]\lambda A=\lambda\left(\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right)=\left(\begin{array}{cc}\lambda a_{11}&\lambda a_{12}\\\lambda a_{21}&\lambda a_{22}\end{array}\right)[/tex]Its trace is [tex]\lambda a_{11}+\lambda a_{22}=\lambda (a_{11}+a_{22})=\lambda \cdot 0=0[/tex]So, [tex]\lambda A\in H[/tex]Therefore, H is a subspace of the vector space V and is closed under addition.