Determinants
Finding the determinants of our matrices is a very useful step that can help us with all kinds of different calculations. But what exactly is a determinant? Why are they so useful, and how can we find them? Let's discover more about this important topic:
Determinants of a matrix
As we may remember, a system of linear equations has three possible outcomes:
- No solutions (the lines run parallel to each other and never intersect)
- Infinite solutions (the functions are equivalent and therefore occupy exactly the same coordinates on a graph)
- One solution( the lines intersect at only one point)
We know that the third possibility is the most common since two random equations are unlikely to be equivalent or share identical slopes (creating parallel lines). Sometimes, we can determine what type of solution our system has just by looking at it. After all, identical slopes and equivalent expressions are relatively easy to stop. But what if it's not so easy?
We can also use determinants to determine the type of solution our system has.
Consider this simple $2\times 2$ square matrix:
$\left[a\phantom{\rule{3pt}{0ex}}b\right]$
$\left[c\phantom{\rule{3pt}{0ex}}d\right]$
The determinant of this square matrix is:
$ad-bc$
In other words, we need to take the difference of the matrix's products. However, we need to multiply our values in a very specific way. The easiest way to explain this is with diagonal lines. The a value is multiplied by the d value first, going from the top left corner to the bottom right corner. Then we multiply the top right value (b) by the bottom left value (c). Finally, we take this second product and subtract it from the first product. No matter how complicated our values get, this always leaves us with the determinant of a 2 by 2 matrix.
Our determinant has a number of interesting and useful characteristics:
- If $ad-bc\ne 0$ , then we know that our linear system of equations has one and only one solution (or point of intersection)
- If $ad-bc=0$ , then we know that our system of linear equations has either no solutions (parallel lines) or infinite solutions (the expressions are equivalent)
We can now see where determinants get their name. By looking at these values, we can determine what kind of solution(s) our system of linear equations might have.
Finding determinants of $3\times 3$ matrices
We can also find the determinants of $3\times 3$ matrices, although the process is a little more complicated. Consider the following $3\times 3$ matrix:
$[{a}_{1}{b}_{1}{c}_{1}]$
$[{a}_{2}{b}_{2}{c}_{2}]$
$[{a}_{3}{b}_{3}{c}_{3}]$
The determinant of this matrix is calculated as follows:
${a}_{1}\times ({b}_{2}{c}_{3}-{c}_{2}{b}_{3})-{b}_{1}\times ({a}_{2}{c}_{3}-{c}_{2}{a}_{3})+{c}_{1}\times ({a}_{2}{b}_{3}-{b}_{2}{a}_{3})$
While this may seem more complex than the $2\times 2$ case, there is a pattern to follow. Each term in the sum involves multiplying one element from the first row by the determinant of the $2\times 2$ submatrix that remains when you eliminate the row and column containing that element. This is often referred to as the minor of the element. The minus signs alternate, which is related to the concept of cofactors in a matrix.
Finding the determinant of a 3x3 matrix involves more than just "taking the difference between diagonal multiples". This process, sometimes called the method of cofactors or the Laplace expansion, involves both addition and subtraction of multiple terms. The process becomes more complex for $4\times 4$ matrices or larger, but it remains an extension of the same principles.
When are determinants useful?
Determinants are useful for a few different operations:
- Solving linear equations (as we have seen above)
- Figuring out how linear transformations change area or volume
- Changing variables in integrals
A determinant can be a useful theoretical tool as well, and it has widespread applications beyond linear equations and matrices. That being said, the concept is most commonly applied to linear equations and matrices.
Determinants are also closely related to Cramer's Rule. This theorem tells us that we can use determinants to find the solutions to a system of linear equations. Not only does Cramer's Rule allow us to solve linear equations, but it also allows us to solve for specific variables in the system. This can be useful if we only need specific variables, and the other variables are unimportant. Once again, we need to know our determinants before we can apply Cramer's rule.
The real question is whether we should use determinants to solve systems of linear equations in the first place. Many mathematicians have correctly pointed out that other methods seem to be more efficient. These methods include Gaussian elimination and QR decompositions to name just a few. We can also determine the type of solution for our linear equation in much simpler ways. For example, we could translate the equation into slope-intercept form and examine the slope. If the slopes are the same, we are looking at parallel lines with no solutions. If the equations are equivalent, then the lines must occupy the same coordinates with infinite solutions. If none of these characteristics are present, there must be only one solution. In most cases, we can make these logical deductions in a few moments -- without using determinants.
With all that said, determinants can be very useful in many situations -- and it's an important concept to learn about as we hone our calculus skills.
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