The determinant is a number calculated from the elements of a square matrix. The determinant of a matrix can be either positive or negative, and is very useful in the solution of systems of linear equations. The determinant is denoted as det(X), det X, or just |X| (the same symbol as in absolute value).
Calculation of the Determinant of a Matrix
Consider the following linear system:
$\begin{cases}
ax+by=A \\
cx+dy=B
\end{cases}$
Let's multiply the first equation by $d$ and the second equation by $-b$:
$\begin{cases}
adx+bdy=dA \\
-cbx-bdy=-bB
\end{cases}$
Now let's add the two equations:
$adx-cbx=dA-bB$
$x=(dA-bB)/(ad-cb)$
Following the same steps, we find that:
$y=(aB-cA)/(ad-cb)$
The term ($ad-cb$) is the determinant of the square matrix $\begin{bmatrix}a & b \\c & d \end{bmatrix}$.
Note that in a 2 X 2 matrix, the determinant is calculated by:
1) multiplying the elements in the main diagonal
2) multiplying the elements in secondary diagonal
3) Subtracting 2) from 1)
For a matrix with 3 rows and 3 columns, such as $\begin{bmatrix}a & b & c\\d & e & f\\g & h & i\end{bmatrix}$, the determinant is calculated like this:
1) Copy the two first columns and paste them beside the matrix (as follows):
$\begin{bmatrix}a & b & c\\d & e & f\\g & h & i\end{bmatrix}$$\begin{bmatrix}a & b\\d & e\\g & h\end{bmatrix}$
2) Sum the products of the three diagonals in the direction north-west to south-east (the direction of the main diagonal):
$aei+bfg+cdh$
3) Sum the products of the three diagonals in the direction south-west to north-east (the direction of the secondary diagonal):
$gec+hfa+idb$
4) Subtract 3) from 2):
Determinant = $(aei+bfg+cdh)-(gec+hfa+idb)$
For example, let's compute the determinant of the square matrix $\begin{bmatrix}2 & 1 & 3\\1 & 1 & 2\\2 & 1 & 1\end{bmatrix}$.
1) Copy the two first columns and paste them beside the matrix:
$\begin{bmatrix}2 & 1 & 3\\1 & 1 & 2\\2 & 1 & 1\end{bmatrix}$$\begin{bmatrix}2 & 1\\1 & 1\\2 & 1\end{bmatrix}$
The determinant is:
$(2*1*1+1*2*2+3*1*1)-(2*1*3+1*2*2+1*1*1)=$
$(2+4+3)-(6+4+1)=9-11=-2$
Minor of an Element
Consider a square matrix A.
$a_{ij}$ is the matrix element in row $i$ and column $j$. The minor ($M_{ij}$) of element $a_{ij}$ is the determinant of the submatrix formed by deleting row $i$ and column $j$ from matrix A.
For example, the Minor of element $a_{13}$ (row 1, column 3) of the matrix $\begin{bmatrix}2 & 1 & 3\\1 & 1 & 2\\2 & 1 & 1\end{bmatrix}$ is the determinant of matrix $\begin{bmatrix}1 & 1\\2 & 1\end{bmatrix}$. That is, $M_{13}=1.1-2.1=-1$.
Cofactor of an Element
Consider a square matrix A.
$a_{ij}$ is the matrix element in row $i$ and column $j$. The Cofactor ($C_{ij}$) of element $a_{ij}$ is computed by multiplying $(-1)^{i+j}$ by the Minor of element $a_{ij}$.
For example, the Cofactor of element $a_{13}$ (row 1, column 3) of the matrix $\begin{bmatrix}2 & 1 & 3\\1 & 1 & 2\\2 & 1 & 1\end{bmatrix}$ is $C_{13}=(-1)^{1+3}*M_{13}=1.(-1)=-1$
Laplace Expansion
According to the Laplace Expansion (or Cofactor Expansion), the ditarminant of a matrix is the sum of the products $a_{ij}(C_{ij})$ for all the elements in a given row or given column.
Let's apply the Laplace Expansion for the following matrix $\begin{bmatrix}2 & 1 & 3\\1 & 1 & 2\\2 & 1 & 1\end{bmatrix}$.
Let's compute the determinant using the elements in the first row:
$a_{11}*C_{11}+a_{12}*C_{12}+a_{13}*C_{13}=$
$2(-1)^{1+1}*D_{11}+1(-1)^{1+2}*D_{12}+3(-1)^{1+3}*D_{13}=$
$2*D_{11}-1*D_{12}+3*D_{13}=$
$2(1*1-1*2)-1(1*1-2*2)+3(1*1-2*1)=$
$2(-1)-1(-3)+3(-1)=-2+3-3=-2$
Now, for comparison, let's compute the determinant using the elements in the third column:
$a_{13}*C_{13}+a_{23}*C_{23}+a_{33}*C_{33}=$
$3(-1)^{1+3}*D_{13}+2(-1)^{2+3}*D_{23}+1(-1)^{3+3}*D_{33}=$
$3*D_{13}-2*D_{23}+1*D_{33}=$
$3(1*1-2*1)-2(2*1-2*1)+1(2*1-1*1)=$
$3(-1)-2(0)+1(1)=-3+0+1=-2$
The determinant will always be the same, no matter what row or what column is chosen.
Properties of the Determinant
1) If all the elements in a row or in a column is equal to zero, the determinant is also zero.
2) If two rows or two columns in a matrix are equal or proportional, the determinant is zero.
3) If all the elements in a given row or in a given column are multiplied by a constant "k", the determinant of the matrix is also multiplied by "k".
4) If two rows or two columns switch position with each other, the determinant of the matrix switches sign.
5) The determinant of the product of two matrices is equal to the product of the determinants of the matrices.
6) The determinant of the matrix $\begin{bmatrix}a & (b+c) & d\\e & (f+g) & h\\i & (j+k) & l\end{bmatrix}$ is equal to the sum of the determinants of the following two matrices: $\begin{bmatrix}a & b & d\\e & f & h\\i & j & l\end{bmatrix}$ and $\begin{bmatrix}a & c & d\\e & g & h\\i & k & l\end{bmatrix}$
In this example we split the second column. But this property applies to any row or any column.
7) Combining properties 3) and 6), we have that the determinant of the matrix $\begin{bmatrix}a & (k.b+c) & d\\e & (k.f+g) & h\\i & (k.j+k) & l\end{bmatrix}$ is equal to the sum of $k$ times the determinant of the matrix $\begin{bmatrix}a & b & d\\e & f & h\\i & j & l\end{bmatrix}$ and the determinant of the matrix $\begin{bmatrix}a & c & d\\e & g & h\\i & k & l\end{bmatrix}$.
In this example we split the second column. But this property applies to any row or any column.
8) If $A^{-1}$ is the the inverse of matrix A, then $det(A^{-1})=\frac{1}{det(A)}$
Solved SAT Practice Tests
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SAT Practice Tests - Determinants of a Matrix and
Additional Practice Tests - Determinants of a Matrix
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