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Mathematical Expression Editor
Finding the Determinant
In this section we will define a function that assigns to each square matrix \(A\) a scalar
output called the determinant of \(A\). We will denote the determinant of \(A\) by \(\det {A}\). For a
matrix with real number entries, the output of the determinant function will always
be a real number.
One important property of the determinant is its connection to matrix inverses. We
will find that a matrix \(A\) is singular if and only if \(\det {A}=0\).
Geometrically speaking, the determinant of a matrix of a linear transformation is
the factor by which the area (or volume or hypervolume) is scaled by the
transformation.
Cofactor Expansion Along the Top Row
To start from the beginning, let us define the determinant of a \(1\times 1\) matrix.
Let \(A=\begin{bmatrix}a\end{bmatrix}\). Define the determinant of \(A\) by \(\det {A}=a\).
It is important to note that this definition is
consistent with our goal of making a connection between determinants and
invertibility. Observe that \(A^{-1}=\begin{bmatrix}a^{-1}\end{bmatrix}\) exists if and only if \(a\neq 0\).
Now we proceed to \(2\times 2\) matrices. According to Formula ??, the inverse of a nonsingular
matrix \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\) is given by
Note the distinction between the square
bracket notation associated with the matrix \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) and the vertical bar notation \(\begin{vmatrix}a&b\\c&d\end{vmatrix}\) used to
denote the determinant in expression (1).
We will now reiterate several important features of this definition and introduce some
vocabulary:
The coefficients \(a\), \(b\) and \(c\) are the entries of the first row of matrix \(A\).
Coefficients in the formula follow an alternating sign pattern: \(+a\), \(-b\), \(+c\). This
pattern will persist in the determinant formulas for determinants of larger
matrices.
When using equation (??), we compute determinants of three matrices:
These matrices are called minor matrices. To form each minor matrix, cross
out the row and column that the corresponding coefficient is in. For example,
the minor matrix corresponding to coefficient \(b\) is found by crossing out the row
and column that \(b\) is in.
The process for finding the determinant described in Definition 6 is referred to
as a cofactor expansion along the top row.
We are starting to observe a certain pattern in the process of computing the
determinant. This pattern will persist for larger matrices. Let’s take a look at a \(4\times 4\)
matrix.
As you watch the video below, pay particular attention to the same patterns as you
saw in the case of \(3\times 3\) matrices: the alternating sign pattern and the process of forming
minor matrices.
We will use the entries in the top row as coefficients in front of \(3\times 3\) determinants. As
before, we will use the alternating sign pattern for the coefficients:
\[+(4), -(-1), +(2), -(1)\]
Just like in the case of a \(3 \times 3\) matrix in Example 4, each of the smaller determinants is
obtained by crossing out the row and the column where the coefficient is located.
We defined the determinant of a matrix in terms of cofactor expansion along the top
row. We will now see what happens when we expand along the first column instead.
We will refer to this process as cofactor expansion along the first column.
Surprisingly, both expansions yield the same result. To illustrate this, let’s revisit
Examples 4 and 11.
In Example 4 we found that \(\det {A}=0\). Let’s try to mimic what we did earlier, but
instead of expanding along the first row, we will expand along the fist column.
In Example 11 we found that \(\det {A}=-58\). We will now try to expand along the fist
column.
When computing determinants of the four \(3\times 3\) matrices below, try different approaches.
You might want to expand along the first row for some of them, and along the first
column for others. Looking for where zeros are located will help you decide what to
try.
In Example 13 and Exploration we were careful not to claim at the outset that we
were finding the determinant of the matrix by cofactor expansion along the first
column; we merely observed that the resulting value was equal to the determinant. It
is possible to prove that both expansions produce the same result (Theorem ??).
Therefore the determinant of a matrix can be defined in terms of cofactor expansion
along the first row or column.
Cofactor Expansion Along Any Row or Column
We originally defined the determinant of a matrix via expansion along the top row of
the matrix. We later observed that expansion along the first column produces the
same result. It turns out that the value of the determinant can be computed by
expanding along any row or column. This result is known as the Laplace Expansion
Theorem (??).
When expanding along an arbitrary row or column, we will continue to follow the two
patterns we observed earlier.
The alternating sign pattern for coefficients will follow the checkerboard
pattern below.
Follow the rules described above to expand along the second row. Compare your
result with the determinant you found in Example 11.
The second row has the advantage over other rows in that it contains a zero. This
will simplify our calculations. Following the checkerboard sign pattern along the
second row we get
This answer is the same as the answer we got using expansion along the first row in
Example 11.
It is clear that having zeros as entries in the matrix reduces the number of
computations necessary to find the determinant. The following example demonstrates
how to use zeros to our advantage.
The fourth column contains the most zeros, so we will expand along that column.
The \((3, 4)\)-entry is the only non-zero entry in the fourth column. Following the
checkerboard pattern, we see that the sign in front of \(-5\) is a minus.
We initially introduced the determinant of a matrix via cofactor expansion along the
top row. We later observed that cofactor expansion along any row or column
produces the same result. We have to be careful, however, not to use a few examples
as “proof" that all cofactor expansions are equivalent. Such claims need to be
carefully supported with general proofs. Unfortunately, in this case, the proofs are
tedious and conceptually unenlightening. An interested reader can find them in
Tedious Proofs Concerning Determinants.
Determinants of Some Special Matrices
We know that we can find the determinant of a matrix by cofactor expansion along
the top row or the first column. (See Theorem ?? of Tedious Proofs Concerning
Determinants for proof.) This property gives rise to a useful result.
Let \(A\) be a square
matrix, then
\[\det {A^T}=\det {A}\]
See Practice Problem .
As we observed earlier, having zeros in a matrix makes it easier for us to compute its
determinant. Recall that that a square matrix is upper-triangular if all of
the entries below the main diagonal are zero. Similarly, a square matrix is
called lower-triangular if all of the entries above the main diagonal are zero.
Together, upper and lower triangular matrices are categorized as triangular
matrices.
If \(A\) is a triangular matrix, then \(\det {A}\) is equal to the product of its diagonal entries.
We proceed by induction on \(n\), where \(A\) is an \(n\times n\) matrix. It is easy to see that this result
holds for \(n=1, 2\). Suppose that the result holds for \((n-1)\times (n-1)\) triangular matrices. We need to show
that it holds for \(n\times n\) triangular matrices.
Suppose \(A=[a_{ij}]\) is a triangular matrix. Then, with the exception of \(a_{11}\), the entries in the first
row (or column) of \(A\) are guaranteed to be zeros. We will take advantage of these zeros
and expand along the first row (or column) of \(A\). As we do so, we obtain a single
product of \(a_{11}\) and the determinant of a minor matrix obtained by crossing out the first
row and column of \(A\). But this minor \((n-1)\times (n-1)\) matrix is also a triangular matrix with diagonal
etries \(a_{22}, a_{33},\ldots , a_{nn}\). By induction hypothesis, its determinant is equal to the product of its
diagonal entries, \(a_{22}\cdot a_{33}\cdot \ldots \cdot a_{nn}\). Therefore
As an immediate consequence of this theorem, we have the following result.
Let \(I\) be the identity matrix, then
\[\det {I}=1\]
We first introduced block matrices in Block Matrix Multiplication. Matrices of the
form \(\begin{bmatrix}A & C\\O& B\end{bmatrix}\) and \(\begin{bmatrix}A & O\\D& B\end{bmatrix}\), where \(A\), \(B\) are square matrices and \(O\) is the zero matrix, are said to be block
triangular. The following theorem makes it easy to compute determinants of such
matrices.
Consider block triangular matrices \(\begin{bmatrix}A & C\\O& B\end{bmatrix}\) and \(\begin{bmatrix}A & O\\D& B\end{bmatrix}\), where \(A\) and \(B\) are square matrices.
Then
Write \(T=\begin{bmatrix}A & C\\O& B\end{bmatrix}\) and proceed by induction on \(k\), where \(A\) is \(k\times k\). If \(k=1\), then the result follows from
cofactor expansion along the first column. In general, let \(S_i(T)\) denote the matrix obtained
from \(T\) by deleting row \(i\) and column 1. Then the cofactor expansion along the first
column is
where \(i=1,2,\dots , k\), \(S_i(A)\) denotes matrix \(A\) with column 1 and row \(i\) deleted, and \(C_i\) denotes matrix \(C\) with
with row \(i\) deleted.
Since \(S_i(A)\) is a \((k-1)\times (k-1)\) matrix, by the induction hypothesis,
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