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Chapter 8: Problem 30

Suppose \(P\) is invertible. Prove that \(\left|P^{-1}\right|=|P|^{-1}\)

Short Answer

Expert verified
We are given that matrix P is invertible and need to prove that \(|P^{-1}| = |P|^{-1}\). We proceed with the following steps: 1. Define the identity matrix and P's inverse: \(P * P^{-1} = I\). 2. Apply the determinant property on product of matrices: \(|P * P^{-1}| = |P||P^{-1}|\). 3. Compute the determinant of the identity matrix: \(|I| = |P||P^{-1}|\). 4. Use the property of identity matrix's determinant (\(|I| = 1\)): \(1 = |P||P^{-1}|\). 5. Solve for \(|P^{-1}|\) by dividing both sides by \(|P|\) (since P is invertible, \(|P|\) cannot be 0): \(|P|^{-1} = |P^{-1}|\). Thus, we have proved that \(|P^{-1}| = |P|^{-1}\).

Step by step solution

01

Define the identity matrix and P's inverse

Recall that for a matrix P, we have an identity matrix I such that P * P^(-1) = I. In other words, multiplying P by its inverse gives us the identity matrix.
02

Apply the determinant property on product of matrices

According to the property |AB|=|A||B|, we have |P * P^(-1)| = |P||P^(-1)|.
03

Compute the determinant of the identity matrix

We know that P * P^(-1) = I. Thus, we get |I| = |P||P^(-1)|.
04

Use the property of identity matrix's determinant

We know that the determinant of the identity matrix is 1, so |I| = 1. Now we have 1 = |P||P^(-1)|.
05

Solve for |P^(-1)|

To prove that |P^(-1)| = |P|^(-1), we need to solve for |P^(-1)|. So, from Step 4, we have: 1 = |P||P^(-1)|. Now, divide both sides by |P| (since P is invertible, |P| cannot be 0): |P|^(-1) = |P^(-1)|. Thus, we proved that |P^(-1)| = |P|^(-1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Invertible Matrix
An invertible matrix is like a key to unlocking the door to solving matrix equations. Also known as a non-singular or non-degenerate matrix, it has some important properties:
  • For a square matrix to be invertible, there must exist another matrix, called the inverse, such that when the two matrices are multiplied, they produce the identity matrix.
  • This is written in the form: if a matrix is denoted by \(P\), then \(P \cdot P^{-1} = I\), where \(I\) is the identity matrix.
  • A fundamental condition for a matrix to be invertible is that its determinant must not be zero. This is because a zero determinant means the matrix doesn't have full rank, indicating it can't be reversed.
Invertibility is crucial because it implies that the matrix can be worked with, manipulated, and used to solve systems of linear equations. Without this property, certain operations cannot be performed.
Identity Matrix
The identity matrix is the magic wand in the kingdom of matrices. It is pivotal for understanding and working with matrix inverses. Here are some key points:
  • An identity matrix is a special kind of square matrix. It has a diagonal of ones and all other elements are zero.
  • Its primary property is leaving other matrices unchanged when multiplied. For any matrix \(A\), \(A \cdot I = A\) and \(I \cdot A = A\).
  • The notation for an identity matrix of size \(n\times n\) is \(I_n\).
Think of the identity matrix as the numeral 1 in regular multiplication. When factored into a product, it doesn't change the other entity. In operations involving inverse matrices, the identity matrix acts as the check that balances the equation, confirming that the inverse is correctly calculated.
Matrix Inverse
Understanding the matrix inverse is vital for engaging effectively with matrix equations. Let's look into its core aspects:
  • The inverse of a matrix \(P\), denoted \(P^{-1}\), is such that multiplying it by \(P\) yields the identity matrix \(I\).
  • Not every matrix has an inverse. A key criterion for existence is a non-zero determinant. Without this, the matrix is deemed non-invertible.
  • The process of finding a matrix's inverse involves specific operations, akin to solving equations using algebra, to rearrange the matrix into a form that reflects its starting condition when multiplied.
The inverse is tremendously useful in matrix algebra for solving systems of equations and transforming equations. It essentially, in the matrix world, does what division does in basic arithmetic.
Determinant Property
The determinant is like the DNA of a matrix鈥攊t contains vital information about its properties. Let's dive into its role and importance:
  • The determinant provides insights into several characteristics of a matrix, such as whether it is invertible or not. If the determinant is zero, the matrix lacks an inverse.
  • One crucial property of determinants is their multiplicative nature: the determinant of a product of matrices equals the product of their determinants, i.e., \(|AB| = |A||B|\).
  • This trait is particularly useful when dealing with inverses as shown in our exercise, where dividing the product of determinants leads to finding the inverse's determinant.
Think of the determinant as a summary of a matrix's characteristics in a single number. It helps determine volume scaling factors and impacts computational efficiency in numeric analysis.

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Most popular questions from this chapter

Use determinants to solve the system \(\left\\{\begin{array}{l}3 y+2 x=z+1 \\\ 3 x+2 z=8-5 y \\ 3 z-1=x-2 y\end{array}\right.\)

Evaluate the determinant of each of the following matrices: (a) $A=\left[\begin{array}{lll}2 & 3 & 4 \\ 5 & 4 & 3 \\ 1 & 2 & 1\end{array}\right]$ (b) $B=\left[\begin{array}{rrr}1 & -2 & 3 \\ 2 & 4 & -1 \\ 1 & 5 & -2\end{array}\right]$ (c) $C=\left[\begin{array}{rrr}1 & 3 & -5 \\ 3 & -1 & 2 \\ 1 & -2 & 1\end{array}\right]\(Use the diagram in Fig. \)8-1$ to obtain the six products: (a) \(|A|=2(4)(1)+3(3)(1)+4(2)(5)-1(4)(4)-2(3)(2)-1(3)(5)=8+9+40-16-12-15=14\) (b) \(|B|=-8+2+30-12+5-8=9\) (c) \(|C|=-1+6+30-5+4-9=25\)

Consider a permutation \(\sigma=j_{1} j_{2} \ldots j_{n} .\) Let \(\left\\{e_{i}\right\\}\) be the usual basis of \(K^{n},\) and let \(A\) be the matrix whose \(i\) th row is \(e_{j_{i}}\left[\text { i.c., } A=\left(e_{j_{1}}, e_{j_{2}}, \ldots, e_{j_{n}}\right)\right] .\) Show that \(|A|=\operatorname{sgn} \sigma\)

$$\operatorname{Let} A=\left[\begin{array}{rrrr} 2 & 1 & -3 & 4 \\ 5 & -4 & 7 & -2 \\ 4 & 0 & 6 & -3 \\ 3 & -2 & 5 & 2 \end{array}\right]$$ (a) Find \(A_{23},\) the cofactor (signed minor) of 7 in \(A\) (b) Find the minor and the signed minor of the submatrix \(M=A(2,4 ; 2,3)\) (c) Find the principal minor determined by the first and third diagonal entries - that is, by \(M=A(1,3 ; \quad 1,3)\)

Compute the determinant of each of the following matrices: (a) \(A=\left[\begin{array}{lll}2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 1\end{array}\right]\) (b) \(B=\left[\begin{array}{rrrr}4 & -6 & 8 & 9 \\ 0 & -2 & 7 & -3 \\ 0 & 0 & 5 & 6 \\ 0 & 0 & 0 & 3\end{array}\right]\) (c) \(\quad C=\left[\begin{array}{rrr}\frac{1}{2} & -1 & -\frac{1}{3} \\\ \frac{3}{4} & \frac{1}{2} & -1 \\ 1 & -4 & 1\end{array}\right]\)
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