Question

a. Find \(b\) if det \(\left[\begin{array}{rrr}5 & -1 & x \\ 2 & 6 & y \\ -5 & 4 & z\end{array}\right]=a x+b y+c z\). b. Find \(c\) if det \(\left[\begin{array}{rrr}2 & x & -1 \\ 1 & y & 3 \\ -3 & z & 4\end{array}\right]=a x+b y+c z\).

Short answer

For part a, \( b = -15 \). For part b, \( c = -7 \).

Step-by-step solution

Recall the Determinant Formula

For a 3x3 matrix \( A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} \), the determinant can be calculated using the formula: \( \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \).

Extract Elements from the Matrix for Part a

For the matrix \( \left[\begin{array}{rrr}5 & -1 & x \ 2 & 6 & y \ -5 & 4 & z\end{array}\right] \), identify the elements: \( a_{11} = 5, \, a_{12} = -1, \, a_{13} = x, \, a_{21} = 2, \, a_{22} = 6, \, a_{23} = y, \, a_{31} = -5, \, a_{32} = 4, \, a_{33} = z \).

Calculate Coefficient of y for Part a

Find the coefficient of \( y \) by determining the minor corresponding to \( a_{23} = y \): the minor is \( \begin{vmatrix} 5 & -1 \ -5 & 4 \end{vmatrix} \). Calculate by using: \( 5 \cdot 4 - (-1)\cdot(-5) = 20 - 5 = 15 \). Thus, \( b = -15 \).

Extract Elements from the Matrix for Part b

For the matrix \( \left[\begin{array}{rrr}2 & x & -1 \ 1 & y & 3 \ -3 & z & 4\end{array}\right] \), identify the elements: \( a_{11} = 2, \, a_{12} = x, \, a_{13} = -1, \, a_{21} = 1, \, a_{22} = y, \, a_{23} = 3, \, a_{31} = -3, \, a_{32} = z, \, a_{33} = 4 \).

Calculate Coefficient of z for Part b

Find the coefficient of \( z \) by determining the minor corresponding to \( a_{32} = z \): the minor is \( \begin{vmatrix} 2 & -1 \ 1 & 3 \end{vmatrix} \). Calculate by: \( 2\cdot3 - (-1)\cdot1 = 6 + 1 = 7 \). Thus, \( c = -7 \).

Key concepts

Matrix Algebra

Matrix algebra is a set of operations that can be performed on matrices, which are rectangular arrays of numbers. These operations include addition, subtraction, and multiplication of matrices, as well as finding the determinant and inverse of a matrix.
Matrices are essential for representing and solving systems of linear equations. In particular, the determinant of a matrix provides valuable information about the matrix. For a 3x3 matrix, the determinant is a scalar value that reflects the volume factor by which the matrix transforms space.
Understanding how to calculate the determinant is a key part of matrix algebra, especially when dealing with transformations and solving equations.

Linear Algebra

Linear algebra is the branch of mathematics concerned with vectors, vector spaces, and linear transformations. It's fundamental in understanding how systems of linear equations can be represented and solved.
One of the core components of linear algebra is the use of matrices to represent linear transformations. A 3x3 matrix can transform vectors in three-dimensional space, and the determinant helps determine properties such as invertibility and the nature of these transformations.
Linear algebra is essential not just in theoretical mathematics but also in fields like engineering, physics, computer science, and more.

Minor of a Matrix

The minor of a matrix is the determinant of a smaller matrix formed by removing one row and one column from the original matrix. For a 3x3 matrix, taking the minor involves creating a 2x2 matrix.
The minor is crucial when calculating the determinant of a larger matrix. For example, in our exercise, calculating the minor of a 2x2 matrix helps in finding coefficients for specific variables in the determinant expression.
This concept is also instrumental in computing cofactors, which in turn are used in finding the inverse of a matrix and understanding its properties.

Coefficient Calculation

Coefficient calculation in matrices involves determining the numerical value that multiplies a specific variable or element within the matrix. In the context of calculating a determinant, this often involves identifying elements that, when multiplied by their corresponding minors, yield the terms of the determinant.
In the example provided, we calculated the coefficients of variables like \( y \) and \( z \) by finding these minors.

• For \( y \), the minor was calculated from a 2x2 section of the original matrix.
• The resulting number (15 in this case) yields the coefficient, which is then adjusted by the sign (leading to \( b = -15 \)).

This method is a staple in matrix algebra for simplifying complex determinants and understanding linear relationships.