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Chapter 4: Problem 39

Find the determinants assuming that \\[\left|\begin{array}{lll}a & b & c \\\d & e & f \\\g & h & i \end{array}\right|=4.\\] \(\left|\begin{array}{lll}2 c & b & a \\ 2 f & e & d \\ 2 i & h & g\end{array}\right|\)

Short Answer

Expert verified
The determinant is 8.

Step by step solution

01

Recognizing the Transformation

Notice that the matrix \( \begin{bmatrix} 2c & b & a \ 2f & e & d \ 2i & h & g \end{bmatrix} \) is obtained from \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) by multiplying the first column by 2.
02

Applying the Scalar Multiplication Rule

According to the properties of determinants, multiplying a column of a matrix by a scalar, here 2, results in multiplying the determinant of the matrix by that scalar. Thus, the determinant of the new matrix is 2 times the original determinant, which is 4.
03

Calculating the New Determinant

Multiply the original determinant (4) by 2, due to the change in the first column.\[ \left|\begin{array}{lll}2c & b & a \ 2f & e & d \ 2i & h & g\end{array}\right| = 2 \times 4 = 8. \]

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Key Concepts

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

Scalar Multiplication Rule
When dealing with matrices, the scalar multiplication rule is a fundamental concept. This rule states that if you multiply a whole column of a matrix by a constant, the determinant of the matrix is also multiplied by that constant. In our example, we changed the original matrix by multiplying the entire first column by 2. Initially, the determinant of the original matrix was given as 4.
Therefore, under the scalar multiplication rule:
  • Identify the column multiplied by the scalar (in this case, the first column).
  • Apply the scalar multiplication: Multiply the determinant by 2.
Thus, the new determinant becomes 8 after the scalar multiplication.
Matrix Transformation
Matrix transformation involves altering a matrix's elements or structure. This can mean simple changes like multiplying a column by a scalar or more complex operations like row interchange. The transformation in the given problem involved multiplying the first column of the matrix by 2. This is a straightforward type of transformation that affects only one part of the matrix.
It's essential to recognize such transformations because they directly influence how you calculate the determinant.
  • For example, multiplying a column changes the determinant, but other transformations like swapping columns would change the sign of the determinant instead.
  • Knowing these effects helps precisely predict and compute the new determinant quickly.
Understanding matrix transformations allows for quick adjustments when computing determinants.
Properties of Determinants
The properties of determinants are crucial tools when working with matrices. They provide shortcuts and rules that simplify otherwise complex calculations. Here are several key properties relevant to determine the outcomes of matrix transformations:
  • Scalar Multiplication: If any matrix column (or row) is multiplied by a scalar, the determinant is also multiplied by that scalar. This was used directly in our example.
  • Row or Column Swap: Swapping two rows or columns changes the sign of the determinant but keeps the magnitude the same.
  • Zero Row or Column: If any row or column consists entirely of zeros, the determinant is 0.
  • Equality: Determinants of identical matrices are equal.
Applying these properties helps in calculating determinants more efficiently and with fewer calculations. In practice, recognizing these properties during transformation simplifies problem-solving and offers quick insights into matrix alterations.

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Most popular questions from this chapter

Draw the Gerschgorin disks for the given matrix. $$\left[\begin{array}{cccc} 2 & \frac{1}{2} & 0 & 0 \\ \frac{1}{4} & 4 & \frac{1}{4} & 0 \\ 0 & \frac{1}{6} & 6 & \frac{1}{6} \\ 0 & 0 & \frac{1}{8} & 8 \end{array}\right]$$

The given matrix is of the form \(A=\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right] .\) In each case, \(A\) can be factored as the product of a scaling matrix and a rotation matrix. Find the scaling factor r and the angle \(\theta\) of rotation. Sketch the first four points of the trajectory for the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\\ 1\end{array}\right]\) and classify the origin as a spiral attractor, spiral repeller, or orbital center. $$A=\left[\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right]$$

Exercise 32 in Section 4.3 demonstrates that every polynomial is (plus or minus) the characteristic polynomial of its own companion matrix. Therefore, the roots of a polynomial p are the eigenvalues of \(C(p)\). Hence, we can use the methods of this section to approximate the roots of any polynomial when exact results are not readily available. In Exercises \(41-44\), apply the shifted inverse power method to the companion matrix \(C(p)\) of \(p\) to approximate the root of \(p\) closest to \(\alpha\) to three decimal places. $$p(x)=x^{3}-5 x^{2}+x+1, \alpha=5$$

Calculate the positive eigenvalue and \(a\) corresponding positive eigenvector of the Leslie matrix \(L\) $$L=\left[\begin{array}{ll} 0 & 2 \\ 0.5 & 0 \end{array}\right]$$

The power method does not converge to the dominant eigenvalue and eigenvector. Verify this, using the given initial vector \(\mathbf{x}_{0}\). Compute the exact eigenvalues and eigenvectors and explain what is happening. $$A=\left[\begin{array}{rrr} -5 & 1 & 7 \\ 0 & 4 & 0 \\ 7 & 1 & -5 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]$$
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