[Linear Algebra] Showing equality via determinant properties

You are using the properties of determinants correctly. In summary, by using the properties of determinants, you can manipulate the given matrix and end up with the desired result on the right hand side. Just remember to divide the determinant by t when you multiply a row or column by t to keep the value unchanged.
  • #1
CentreShifter
24
0
Problem:

Show, without evaluating directly, that
[tex]
\left|\begin{matrix}
a_1+b_1t&a_2+b_2t&a_3+b_3t \\
a_1t+b_1&a_2t+b_2&a_3t+b_3 \\
c_1&c_2&c_3 \end{matrix}\right|
=
(1-t^2)\left|\begin{matrix}
a_1&a_2&a_3 \\
b_1&b_2&b_3 \\
c_1&c_2&c_3 \end{matrix}\right|
[/tex]

Clearly, here I'm supposed to use the determinant properties, do some row ops on the first array, and end up with the RHS.

1. -tR1-R2 -> Row2 (no coefficient on determinant).

[tex]
\left|\begin{matrix}
a_1+b_1t&a_2+b_2t&a_3+b_3t \\
b_1-b_1t^2&b_2-b_2t^2&b_3-b_3t^2 \\
c_1&c_2&c_3 \end{matrix}\right|
[/tex]

From here I can see that [tex]b_1-b_1t^2=b_1(1-t^2)[/tex]. But multiplying R2 by [tex]\frac{1}{1-t^2}[/tex] means I have to also pull that out as a coefficient to the entire array. So now:
[tex]
\frac{1}{1-t^2}
\left|\begin{matrix}
a_1+b_1t&a_2+b_2t&a_3+b_3t \\
b_1&b_2&b_3 \\
c_1&c_2&c_3 \end{matrix}\right|
[/tex]

Another row op, R1 -> R1-tR2, and I have my RHS, except the coefficient is reciprocated. Am I doing this wrong?
 
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  • #2
If you multiply a row or column by some t, the value of the determinant becomes t times the original value. So you need to divide the whole determinant by t so as to keep it unchanged.

ehild
 

1. What are the properties of determinants?

The properties of determinants include: multiplication by a scalar, addition of rows or columns, switching rows or columns, and the determinant of a triangular matrix is equal to the product of its diagonal elements.

2. How do I use the properties of determinants to show equality?

You can use the properties of determinants to manipulate and simplify both sides of the equation until they are equal. This can be done by applying the properties in a step-by-step manner.

3. Can I use any of the properties of determinants to show equality?

Yes, you can use any of the properties of determinants to show equality. However, it is important to keep in mind that some properties may be more useful in certain situations than others.

4. Are there any special cases where the properties of determinants may not work?

Yes, there are some special cases where the properties of determinants may not work. For example, if the matrix is not square, the properties may not apply. Additionally, if the matrix has a zero determinant, the properties may not be useful in showing equality.

5. Can the properties of determinants be used to prove other mathematical concepts?

Yes, the properties of determinants can be used to prove other mathematical concepts. For example, they can be used to prove the invertibility of a matrix or to find the inverse of a matrix.

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