Question

Matrices \(A\) and \(\vec{b}\) are given. (a) Give \(\operatorname{det}(A)\) and \(\operatorname{det}\left(A_{i}\right)\) for all \(i\). (b) Use Cramer's Rule to solve \(A \vec{x}=\vec{b}\). If Cramer's Rule cannot be used to find the solution, then state whether or not a solution exists. \(A=\left[\begin{array}{ccc}1 & 0 & -10 \\ 4 & -3 & -10 \\ -9 & 6 & -2\end{array}\right]\) \(\vec{b}=\left[\begin{array}{c}-40 \\ -94 \\ 132\end{array}\right]\)

Short answer

Det(A) = 36; Use Cramer's Rule to solve since Det(A) ≠ 0. Calculate Dets Ai and solve.

Step-by-step solution

Calculate determinant of A

The determinant of matrix \(A\) is computed as follows: \[ \det(A) = \begin{vmatrix} 1 & 0 & -10 \ 4 & -3 & -10 \ -9 & 6 & -2 \end{vmatrix} \]Using the formula for determinants of 3x3 matrices: \[ \det(A) = 1((-3) \cdot (-2) - (-10) \cdot 6) - 0(4 \cdot (-2) - (-10) \cdot (-9)) + (-10)(4 \cdot 6 - (-3) \cdot (-9)) \]Calculate each term: \[ \det(A) = 1(6 + 60) + (-10)(24 - 27) = 66 - 30 = 36 \] Thus, \( \det(A) = 36 \).

Define matrices Ai

To use Cramer's Rule, define matrices \(A_i\), which are derived from \(A\) by replacing the \(i^{th}\) column with vector \(\vec{b}\). We need to compute \(\det(A_i)\) for each \(i = 1, 2, 3\). \(A_1 = \begin{bmatrix} -40 & 0 & -10 \ -94 & -3 & -10 \ 132 & 6 & -2 \end{bmatrix} \)\(A_2 = \begin{bmatrix} 1 & -40 & -10 \ 4 & -94 & -10 \ -9 & 132 & -2 \end{bmatrix} \)\(A_3 = \begin{bmatrix} 1 & 0 & -40 \ 4 & -3 & -94 \ -9 & 6 & 132 \end{bmatrix} \)

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number calculated from a square matrix. It offers important information about the matrix itself, including whether it is invertible. For a 3x3 matrix, the formula to calculate the determinant involves a combination of multiplication and subtraction of elements from the matrix.

In general, the determinant of a 3x3 matrix:- Assume the matrix is \[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]- The determinant is calculated as follows:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]To solve linear equations using Cramer's Rule, calculating the determinant is essential, as Cramer's Rule requires that the determinant of the main matrix \(A\) is non-zero. If \(\det(A) = 0\), Cramer's Rule can't be applied, indicating either no solutions or infinitely many solutions exist for the system.

Solving Linear Systems

Solving linear systems is fundamental in mathematics and involves finding the values of variables that satisfy a set of linear equations. Cramer's Rule is a powerful method used to solve such systems, but it is applicable only when the matrix \(A\) is invertible (i.e., \(\det(A) eq 0\)).

Cramer's Rule states that if you have a system of equations represented in matrix form as \(A\vec{x} = \vec{b}\), and \(A\) is a square matrix with a non-zero determinant, then each variable \(x_i\) can be solved as:- \(x_i = \frac{\det(A_i)}{\det(A)}\)where \(A_i\) is the matrix obtained by replacing the \(i^{th}\) column of \(A\) with the vector \(\vec{b}\).The rule provides a straightforward yet powerful solution method for small systems, but it can be cumbersome for larger matrices. It is also essential to calculate the determinants accurately to ensure correct solutions are derived.

Matrix Algebra

Matrix algebra is a set of rules and operations designed to work with matrices, which are rectangular arrays of numbers. This form of algebra includes operations such as addition, subtraction, and multiplication of matrices, as well as calculating determinants and inverses.

- **Addition/Subtraction:** Two matrices can be added or subtracted if they have the same dimensions. This is done by adding or subtracting the corresponding elements. - **Multiplication:** Multiplying matrices involves the dot product of the rows of the first matrix with the columns of the second. This operation is only defined when the number of columns in the first matrix matches the number of rows in the second. - **Determinants and Inverses:** As seen with Cramer's Rule, determinants are crucial in determining the invertibility of a matrix. An invertible matrix has an inverse, which is a matrix that when multiplied by the original yields the identity matrix. Matrix algebra forms the basis of solving complex linear systems, allowing for operations that would be cumbersome if attempted with plain algebra, and providing tools necessary for applications in physics, engineering, and computer science.