Question

if A is a matrix where A= 1 2 2 1 and f(x)=x²-2x-3. show that f(A)=0.

Answer 1

Solution. The answer is that only (c) and (d) have any chance of being equal,
since they are the only matrices in the list with the same size (2 × 2). As a
matter of fact, an entry-by-entry check verifies that they really are equal. -

Matrix Addition and Subtraction
How should we define addition or subtraction of matrices? We take a clue
from elementary two- and three-dimensional vectors, such as the type we
would encounter in geometry or calculus. There, in order to add two vectors,
one condition has to hold: the vectors have to be the same size. If they are
the same size, we simply add the vectors coordinate by coordinate to obtain
a new vector of the same size. That is precisely what the following definition
does.
Matrix Definition 2.2. Let A = [aij ] and B = [bij ] be m × n matrices. Then the
Addition and
Subtraction
sum of the matrices, denoted by A + B, is the m × n matrix defined by the
formula
A + B = [aij + bij ] .
The negative of the matrix A, denoted by −A, is defined by the formula
−A = [−aij ] .
Finally, the difference of A and B, denoted by A−B, is defined by the formula
A − B = [aij − bij ] .
Notice that matrices must be the same size before we attempt to add them.
We say that two such matrices or vectors are conformable for addition.
Example 2.2. Let
A =
310
−201 -

and B =

−321
140 -

.
Find A + B, A − B, and −A.
Solution. Here we see that
A + B =
310
−201 -

+

−321
140 -

=
3 − 3 1+20+1
−2+10+41+0 -

=
031
−141 -

.
Likewise,
A − B =
310
−201 -

−

−321
140 -

=

3 − −3 1 − 2 0 − 1
−2 − 1 0 − 4 1 − 0
-

=
6 −1 −1
−3 −4 1 -

.
The negative of A is even simpler:
−A =
−3 −1 −0
− − 2 −0 −1
-

=

−3 −1 0
2 0 −1
-

.