Answer:
See attached pictures.
Step-by-step explanation:
See attached pictures.
(a) The set V of all 3×3 matrices is a lower triangular matrix.
(b) It is found that V is not a group under addition. Hence V is not a vector space.
Matrix of Lower-Triangular Shapes A lower-triangular matrix is one that contains nonzero entries solely on the diagonal and below it.
(a) Let V is a 3×3 lower triangular matrix
If in these cases V belongs to the null matrices as;
[tex]\left[\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right][/tex]
A∈V, B∈V, A-B∈V
[tex]\rm A =\left[\begin{array}{ccc}a_1&0&0\\a_2&a_3&0\\a_4&a_5&a_6\end{array}\right][/tex]
[tex]\rm B =\left[\begin{array}{ccc}b_1&0&0\\b_2&b_3&0\\b_4&b_5&b_6\end{array}\right][/tex]
[tex]\rm A-B =\left[\begin{array}{ccc}a_1-b_1&0&0\\a_2-b_2&a_3-b_3&0\\a_4-b_4&a_5-b_5&a_6-b_6\end{array}\right][/tex]
AB is a lower triangular matrix.
(b) V is a vector space with a 2×2 matrix with the determinant is equal to zero.
[tex]\rm A =\left[\begin{array}{cc}1&1\\1&1&\end{array}\right][/tex]
[tex]\rm B= \left[\begin{array}{cc}1&-1\\-1&1&\end{array}\right][/tex]
det (A)= 1-1 = 0
det (B)=(-1)-(-1)= 0
[tex]\rm A+B =\left[\begin{array}{cc}2&0\\0&2&\end{array}\right][/tex]
det (A+B)= 4
A+B ∉V
It is found that V is not a group under addition . Hence V is not a vector space.
To learn more about the lower triangular matrix refer to the link;
https://brainly.com/question/15280480
