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Chapter 8: Problem 12

Find the given minor and cofactor pertaining to the matrix $$\left[\begin{array}{rrr} -3 & 0 & 2 \\ 1 & 5 & -4 \\ 0 & 6 & 5 \end{array}\right]$$ $$M_{21} \text { and } C_{21}$$

Short Answer

Expert verified
The minor \(M_{21}\) is -12 and the cofactor \(C_{21}\) is 12.

Step by step solution

01

Calculate the minor

To find the minor \(M_{21}\), the second row and first column are removed from the matrix. This results in a 2x2 matrix: \(\begin{bmatrix}0&2\6&5\end{bmatrix}\). The minor \(M_{21}\) is worked out as the determinant of this 2x2 matrix which is equal to (0)(5) - (2)(6).
02

Simplify the minor

Simplify the obtained result to get the value of the minor \(M_{21} = 0*5 - 2*6 = -12\). This obtained value represents the minor of the element located in row 2 and column 1 of the original 3x3 matrix.
03

Calculate the cofactor

The cofactor \(C_{21}\) is calculated using the formula \(C_{21} = (-1)^{2+1} * M_{21}\). Substituting the calculated value of \(M_{21} = -12\) into this equation will give \(C_{21}\).
04

Simplify the cofactor

To simplify \(C_{21}\) value, calculate \(C_{21} = -1 * -12 = 12\). So, the cofactor \(C_{21} = 12\).

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Key Concepts

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

Minor of a Matrix
Understanding the minor of a matrix is essential in matrix algebra. A minor of a matrix is derived by removing a row and a column from a given matrix to form a smaller matrix. In essence, it's the determinant of this resulting smaller or sub-matrix. This is a key step when calculating cofactors and determinants of larger matrices.
In our original exercise, we are tasked to find the minor \( M_{21} \) of the matrix:
  • First, identify the element at row 2, column 1.
  • Remove row 2 and column 1, leaving a 2x2 matrix: \( \begin{bmatrix}0 & 2 \ 6 & 5\end{bmatrix} \).
  • Calculate the determinant of this 2x2 matrix: \( (0 \cdot 5) - (2 \cdot 6) = 0 - 12 = -12 \).
The minor \( M_{21} \) for element at row 2 and column 1 is \(-12\). This value plays an important role in calculating cofactors and understanding the matrix structure.
Cofactor of a Matrix
The cofactor of a matrix builds on the minor by introducing a sign factor, which arises from the position of the element within the original matrix. The formula for a cofactor is given by:
  • \( C_{ij} = (-1)^{i+j} \cdot M_{ij} \)
  • The sign is "+" if the sum of the indices \( i + j \) is even and "-" if it is odd.
For instance, solving for \( C_{21} \):
  • In the original exercise, \( i = 2 \) and \( j = 1 \), making \( i + j = 3 \). Since 3 is odd, we insert a negative sign.
  • The cofactor formula becomes \( C_{21} = (-1)^{3} \cdot (-12) \).
  • Calculate \( C_{21} = -1 \times -12 = 12 \).
The cofactor \( C_{21} \) is used in determining the determinant of larger matrices and is crucial for matrix inversion.
Determinant Calculation
Calculating the determinant is pivotal in linear algebra, serving as a tool to determine matrix properties like invertibility. The determinant provides insight into the linear transformations represented by a matrix.
For a 2x2 matrix:
  • A simple formula is used, \( \text{det} = ad - bc \) for a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
For larger matrices, like 3x3, determinants are often found through expansion by minors and cofactors:
  • Choose a row or column (often the first row or column for simplicity).
  • Sum the products of each element and their corresponding cofactors.
  • For the given matrix, if the determinant calculation utilized the first row, it would involve elements and their cofactors along that row.
Understanding these concepts helps in solving systems of equations, especially in finding unique solutions, and examining the matrix's rank.

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Most popular questions from this chapter

The area of a rectangular property is 300 square feet. Its length is three times its width. There is a rectangular swimming pool centered within the property. The dimensions of the property are twice the corresponding dimensions of the pool. The portion of the property that lies outside the pool is paved with concrete. What are the dimensions of the property and of the pool? What is the area of the paved portion?

Keith and two of his friends, Sam and Cody, take advantage of a sidewalk sale at a shopping mall. Their purchases are summarized in the following table. $$\begin{array}{lc|c|c|} \hline& {3}{|}\text { Quantity } \\\\\hline\text { Name } & \text { Shirt } & \text { Sweater } & \text { Jacket } \\\\\hline \text { Keith } & 3 & 2 & 1 \\\\\text { Sam } & 1 & 2 & 2 \\\\\text { Cody } & 2 & 1 & 2\\\\\hline\end{array}$$ The sale prices are \(\$ 14.95\) per shirt, \(\$ 18.95\) per sweater, and \(\$ 24.95\) per jacket. In their state, there is no sales tax on purchases of clothing. Use matrix multiplication to determine the total expenditure of each of the three shoppers.

Consider the following augmented matrix. For what value(s) of \(a\) does the corresponding system of linear equations have infinitely many solutions? One solution? Explain your answers.$$\left[\begin{array}{lll|r}1 & 0 & 0 & -2 \\\0 & 1 & 0 & 5 \\\0 & 0 & a & 0\end{array}\right]$$.

Consider the following system of equations. $$\left\\{\begin{aligned} x^{2}+y^{2} &=r^{2} \\ (x-h)^{2}+y^{2} &=r^{2} \end{aligned}\right.$$ Let \(r\) be a (fixed) positive number. For what value(s) of \(h\) does this system have (a) exactly one real solution? (b) exactly two real solutions? (c) infinitely many real solutions? (d) no real solution? (Hint: Visualize the graphs of the two equations.)

Let \(I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{rr}2 & -1 \\ 1 & 0\end{array}\right] .\) Calculate \(A I\) and IA. What do you observe?
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