18.03 LA.4: Inverses and Determinants
[1] Transposes
[2] Inverses
[3] Determinants
[1] Transposes
The transpose of a matrix A is denoted AT , or in Matlab, A0 .
The transpose of a matrix exchanges the rows and columns. The ith
column becomes the ith row. Or the aij entry becomes the aji entry.
Example:
1 3
1 2 4 2 5
A=
3 5 7
4 7
Symmetric Matrices are square matrices that satisfy A = AT .
Example:
1 3 9
A = 3 2 5
9 5 8
We’ll see that the eigenvalues of symmetric matrices are great. The eigen-
vectors are even better! And symmetric matrices come up all of the time.
Property of transposes:
(AB)T = B T AT
1
, [2] Inverses
Important questions:
• Is a matrix A invertible?
• How do you compute the inverse?
Let A be a square matrix. Suppose it has an inverse. We denote the
inverse by A−1 , and it has the property that
AA−1 = I A−1 A = I.
The fact that the inverse is simultaneously a right and left inverse is not
immediately obvious. See if you can use the associative property (AB)C =
A(BC) to see why this must be the case when A is square.
If the inverse of A and B both exists, and both matrices have the same
shape, then
(AB)−1 = B −1 A−1
Corny Example:
If B represents taking off your jacket, and A represents taking off your
sweater, then it makes sense that you first take of your jacket, and then
take off your sweater. To find the inverse, which is to reverse this process,
it makes sense that we have to reverse the order. First you put the sweater
back on, and then you put your jacket on.
So let’s start to answer our question: when is a matrix invertible? To
answer this question, we’ll look at when it is NOT invertible first.
A is NOT invertible when:
• The determinant of A is zero.
• There exists a nonzero vector x so that Ax = 0.
2
[1] Transposes
[2] Inverses
[3] Determinants
[1] Transposes
The transpose of a matrix A is denoted AT , or in Matlab, A0 .
The transpose of a matrix exchanges the rows and columns. The ith
column becomes the ith row. Or the aij entry becomes the aji entry.
Example:
1 3
1 2 4 2 5
A=
3 5 7
4 7
Symmetric Matrices are square matrices that satisfy A = AT .
Example:
1 3 9
A = 3 2 5
9 5 8
We’ll see that the eigenvalues of symmetric matrices are great. The eigen-
vectors are even better! And symmetric matrices come up all of the time.
Property of transposes:
(AB)T = B T AT
1
, [2] Inverses
Important questions:
• Is a matrix A invertible?
• How do you compute the inverse?
Let A be a square matrix. Suppose it has an inverse. We denote the
inverse by A−1 , and it has the property that
AA−1 = I A−1 A = I.
The fact that the inverse is simultaneously a right and left inverse is not
immediately obvious. See if you can use the associative property (AB)C =
A(BC) to see why this must be the case when A is square.
If the inverse of A and B both exists, and both matrices have the same
shape, then
(AB)−1 = B −1 A−1
Corny Example:
If B represents taking off your jacket, and A represents taking off your
sweater, then it makes sense that you first take of your jacket, and then
take off your sweater. To find the inverse, which is to reverse this process,
it makes sense that we have to reverse the order. First you put the sweater
back on, and then you put your jacket on.
So let’s start to answer our question: when is a matrix invertible? To
answer this question, we’ll look at when it is NOT invertible first.
A is NOT invertible when:
• The determinant of A is zero.
• There exists a nonzero vector x so that Ax = 0.
2
