We want to find the value of A
that makes matrix A singular.
Notice a is the entry
in row one, column one.
A square matrix A is singular
if it does not have an inverse matrix.
We learned before that
matrix A is invertible
or nonsingular if the
determinant does not equal zero,
which means matrix A is singular
if the determinant of A does equal zero.
So we'll find the determinant of matrix A,
set it equal to zero,
and then solve for a.
So we can use vertical bars
to represent the determinant.
If we use expansion by
minors in the first row,
we can use this formula here
to evaluate the determinant.
So the first element in row one is a.
So we'll have a times a
two-by-two determinant
where to find these elements
we eliminate the row
and column of element a.
So we eliminate row one, column one.
The remaining elements give us
this two by two determinant.
So we have two, three, one, two,
and we have minus the
next element in row one
is negative one.
So we have minus and then negative one
times a two-by-two determinant.
We're to find the elements' determinant,
we eliminate the row and
column of negative one,
which would be row one, column two,
which gives us three, three, two, two,
then we have plus the
last element in row one
is negative three, so I
have plus negative three
times another two-by-two determinant,
to find the elements in this determinant
we eliminate the row and
column of negative three
which would be row one, column three.
So we have three, two, two, one,
and this must be equal to zero if we want
matrix A to be singular.
And now to evaluate each
two-by-two determinant,
we'll find this product
minus this product.
So here we have a times, this
will be four minus three,
this becomes plus one times,
this is six minus six,
then we have plus negative
three, that's with that as
minus three times, here
we have three minus four
equals zero, so
simplifying again we have a
plus this is one times zero, that's zero,
and here we have negative
three times negative one
which is positive three equals zero,
subtracting three on both sides,
a equals negative three
if matrix A is singular.
I hope you found this helpful.
