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Chapter 11: Problem 20

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don鈥檛 calculate the inverse. $$ \left[\begin{array}{rrr}{1} & {2} & {5} \\ {-2} & {-3} & {2} \\ {3} & {5} & {3}\end{array}\right] $$

Short Answer

Expert verified
The matrix has no inverse since its determinant is 0.

Step by step solution

01

Write down the formula for the determinant of a 3x3 matrix

To find the determinant of a 3x3 matrix \( A \) with elements \( a_{ij} \), the formula is given by:\[det(A) = a_{11}(a_{22}a_{33} - a_{32}a_{23}) - a_{12}(a_{21}a_{33} - a_{31}a_{23}) + a_{13}(a_{21}a_{32} - a_{31}a_{22})\]
02

Identify the elements from the matrix

For the matrix \[\begin{bmatrix}1 & 2 & 5 \-2 & -3 & 2 \3 & 5 & 3\end{bmatrix}\]The elements are: \( a_{11} = 1, a_{12} = 2, a_{13} = 5, a_{21} = -2, a_{22} = -3, a_{23} = 2, a_{31} = 3, a_{32} = 5, a_{33} = 3 \).
03

Substitute the elements into the determinant formula

Substitute each element into the determinant formula:\[det(A) = 1((-3)(3) - (5)(2)) - 2((-2)(3) - (3)(2)) + 5((-2)(5) - (3)(-3))\]
04

Calculate the individual products

Calculate the products:\((-3)(3) = -9\), \((5)(2) = 10\)\((-2)(3) = -6\), \((3)(2) = 6\)\((-2)(5) = -10\), \((3)(-3) = -9\)
05

Simplify the determinant equation

Simplify using the calculated products:\[det(A) = 1(-9 - 10) - 2(-6 - 6) + 5(-10 + 9)\]\[det(A) = 1(-19) - 2(-12) + 5(-1)\]\[det(A) = -19 + 24 - 5\]
06

Compute the final determination value

Sum the values obtained:\[det(A) = -19 + 24 - 5 = 0\]
07

Determine if the matrix has an inverse

A matrix has an inverse if and only if its determinant is non-zero. Since the determinant of the given matrix is 0, it does not have an inverse.

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Key Concepts

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

Inverse of a Matrix
Understanding the concept of the inverse of a matrix is crucial in linear algebra. Let's imagine a matrix as a function that transforms one vector into another. The inverse matrix essentially "undoes" this transformation, taking you back to the original vector.
  • If a matrix of size 3x3 has an inverse, multiplying the matrix by its inverse yields the identity matrix. The identity matrix acts like the number 1 in scalar arithmetic.
  • For any matrix \( A \), its inverse is denoted as \( A^{-1} \).
  • An important rule is that a matrix must have a non-zero determinant for it to have an inverse.
In our exercise, we found that the determinant is zero. This means the matrix does not have an inverse. The absence of an inverse suggests that its rows or columns are linearly dependent, meaning one can be written as a combination of others.
3x3 Matrix Determinant Formula
The determinant of a matrix is a special number that can be calculated from its elements. For a 3x3 matrix, it helps us understand a few key properties, including whether the matrix has an inverse.
To calculate the determinant of a 3x3 matrix, we use the formula: \[\det(A) = a_{11}(a_{22}a_{33} - a_{32}a_{23}) - a_{12}(a_{21}a_{33} - a_{31}a_{23}) + a_{13}(a_{21}a_{32} - a_{31}a_{22})\]This formula might look complex, but it's essentially breaking the matrix into smaller 2x2 determinants. Each term corresponds to a row element of the first row and is involved in multiplying and subtracting smaller determinants formed from the matrix.
  • Understanding the pattern can simplify remembering the formula: it's alternately adding and subtracting 2x2 determinants.
  • Each of these smaller 2x2 determinants involves a pair of elements from the second and third rows.
Determinant Calculation Steps
Calculating the determinant of a 3x3 matrix involves several well-defined steps. For any given matrix, you should first write out the detailed formula for the determinant.
  • **Step 1:** Identify each element of the matrix and understand where it fits into the formula.
  • **Step 2:** Substitute these elements into the determinant equation. Each part of the formula corresponds to a specific combination of matrix elements.
  • **Step 3:** Calculate the products for each part of the formula. It involves pairing elements smartly and performing basic multiplication.
  • **Step 4:** Simplify the expression by combining these results. Pay close attention to the signs as you simplify computational steps.
  • **Step 5:** Summarize to get the determinant value.
In the example provided, simplifying and calculating these steps correctly led us to a final determinant of zero, indicating the matrix has no inverse. It's important to execute each step methodically to prevent errors, especially in sign handling and arithmetic.

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Most popular questions from this chapter

Use Cramer鈥檚 Rule to solve the system. $$ \left\\{\begin{aligned} x-y+2 z =0 \\ 3 x \quad\quad+z=11 \\\\-x+2 y\quad\quad =0 \end{aligned}\right. $$

Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{ll}{2} & {0} \\ {0} & {3}\end{array}\right] $$

Use Cramer鈥檚 Rule to solve the system. $$ \left\\{\begin{aligned} 2 x-y &=-9 \\ x+2 y &=8 \end{aligned}\right. $$

Shading Unwanted Regions To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? Why? $$ \left\\{\begin{aligned} x+2 y &>4 \\\\-x+y &<1 \\ x+3 y &<9 \\ x &<3 \end{aligned}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don鈥檛 calculate the inverse. $$ \left[\begin{array}{rrr}{2} & {1} & {0} \\ {0} & {-2} & {4} \\ {0} & {1} & {-3}\end{array}\right] $$
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