The Determinant of a Matrix
In the world of linear algebra, a branch of mathematics concerned with vectors, vector spaces, and linear transformations, the determinant of a matrix is a key concept. If this sounds a bit complex, worry not, we will break it down in a straightforward manner.
So, what exactly is a determinant?
Visualize a square matrix, essentially an orderly set of numbers arrayed in rows and columns.
$$ M = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$
The determinant is a value derived from this matrix and it can provide a great deal of information about the matrix itself.
It's represented either by the uppercase delta symbol Δ or by the matrix's name flanked by two vertical lines.
$$ \Delta_M = | M | $$
Remember, a determinant isn't merely a number - it's a tool helping us better understand the matrix's properties.
Why is this important? Simply put, the determinant aids us in solving a plethora of mathematical problems. It allows us to compute solutions for systems of linear equations, understand shape and volume transformations in space, and so much more.
How is the Determinant Calculated?
The way to calculate the determinant can vary depending on the size of the matrix.
For smaller matrices, such as 2x2 or 3x3 ones, handy formulas exist to simplify the computation.
For larger matrices, however, more complex methods are used, like Laplace's expansion or Gauss's algorithm.
For 2x2 Matrices
If you have a 2x2 matrix, with two rows and two columns
$$ M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$
you can calculate the determinant using the formula ad - bc
$$ \Delta_M = a \cdot d - b \cdot c $$
You essentially calculate the product of the elements on the main diagonal, then subtract the product of the elements on the secondary diagonal.
Consider a 2x2 matrix, for example, $$ M = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$ In this case, the elements are a=1, b=2, c=3, and d=4. Apply the above formula to calculate the determinant of the matrix $$ \Delta_M = a \cdot d - b \cdot c = 1 \cdot 4 - 2 \cdot 3 = 4 - 6 = -2 $$ The determinant of the matrix is ΔM=-2.
For 3x3 Matrices
In the case of a 3x3 matrix, with three rows and three columns
$$ M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} $$
you can use Sarrus' rule.
$$ \Delta = aei + bfg + cdh - gec - hfa - idb $$
In essence, you append the first two columns to the end of the matrix.
$$ M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{matrix} a & b \\ d & e \\ g & h \end{matrix} $$
Then, you need to add the products of the main diagonals aei + bfg + cdh
$$ M = \begin{pmatrix} \color{blue}a & \color{red}b & \color{green}c \\ d & \color{blue}e & \color{red}f \\ g & h & \color{blue}i \end{pmatrix} \begin{matrix} a & b \\ \color{green}d & e \\ \color{red}g & \color{green}h \end{matrix} $$
Finally, you subtract those of the secondary diagonals - gec - hfa - idb
$$ M = \begin{pmatrix} a & b & \color{blue}c \\ d & \color{blue}e & \color{red}f \\ \color{blue}g & \color{red}h & \color{green}i \end{pmatrix} \begin{matrix} \color{red}a & \color{green}b \\ \color{green}d & e \\ g & h \end{matrix} $$
Example. Consider this 3x3 matrix $$ M = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} $$ In this case, the elements are a=1, b=2, c=3, d=4, e=5, f=6, g=7, h=8, i=9. Apply Sarrus' rule to calculate the determinant of the matrix $$ \Delta_M = aei + bfg + cdh - gec - hfa - idb $$ $$ \Delta_M = 1 \cdot 5 \cdot 9 + 2 \cdot 6 \cdot 7 + 3 \cdot 4 \cdot 8 - 7 \cdot 5 \cdot 3 - 8 \cdot 6 \cdot 1 - 9 \cdot 4 \cdot 2 $$ $$ \Delta_M = 45 + 84 + 96 - 105 - 48 - 72 $$ $$ \Delta_M = 0 $$ Hence, the determinant of the matrix M is ΔM=0.
Properties of the Determinant
The determinant possesses some highly useful properties.
Property 1
For instance, if all the numbers in a row or a column of the matrix are zero, then the determinant is also zero.
Consider the matrix $$ M = \begin{pmatrix} 1 & 2 & 0 \\ 4 & 5 & 0 \\ 7 & 8 & 0 \end{pmatrix} $$ In this case, as a column consists entirely of zeros, the determinant is certainly zero $$ \Delta_M = 0 $$
Property 2
If two rows or columns are identical or proportional, the determinant is also zero
For example, in the matrix $$ M = \begin{pmatrix} 1 & 2 & 4 \\ 4 & 5 & 10 \\ 7 & 8 & 16 \end{pmatrix} $$ the third column is exactly double the second one. Here again, we can confidently assert that the determinant is zero $$ \Delta_M = 0 $$
Property 3
Moreover, the determinant has a multiplicative property: the determinant of the product of two matrices equals the product of their determinants.
For instance, consider two matrices $$ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$ $$ B = \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix} $$ The determinants of the matrices are $$ \Delta_A = -2 $$ $$ \Delta_B = 3 $$ The product of the two determinants equals -6 $$ \Delta_A \cdot \Delta_B = -6 $$ When the two matrices are multiplied together A·B $$ A \cdot B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 5 & 6 \\ 11 & 12 \end{pmatrix} $$ The determinant of the product matrix A·B is -6 $$ \Delta_{A·B} = 5 \cdot 12 - 6 \cdot 11 = 60 - 66 = -6 $$ The determinant of the product matrix det(A·B) equals the product of the determinants det(A) and det(B)
Whether you're studying or working in a field that harnesses linear algebra - such as engineering, physics, economics, or computer science - understanding determinants is imperative.
Though it may seem somewhat intricate at first, with some practice, calculating determinants becomes quite straightforward.
