Question

Use determinants to find which real values of \(c\) make each of the following matrices invertible. a. \(\left[\begin{array}{rrr}1 & 0 & 3 \\ 3 & -4 & c \\ 2 & 5 & 8\end{array}\right]\) b. \(\left[\begin{array}{rrr}0 & c & -c \\ -1 & 2 & 1 \\ c & -c & c\end{array}\right]\) c. \(\left[\begin{array}{rrr}c & 1 & 0 \\ 0 & 2 & c \\ -1 & c & 5\end{array}\right]\) d. \(\left[\begin{array}{lll}4 & c & 3 \\ c & 2 & c \\ 5 & c & 4\end{array}\right]\) e. \(\left[\begin{array}{rrr}1 & 2 & -1 \\ 0 & -1 & c \\ 2 & c & 1\end{array}\right]\) f. \(\left[\begin{array}{rrr}1 & c & -1 \\ c & 1 & 1 \\ 0 & 1 & c\end{array}\right]\)

Short answer

a) \(c \neq \frac{19}{5}\); b) \(c \neq 0\); c) \(c \neq 0\); d-e) any \(c\); f) any \(c\).

Step-by-step solution

Understand Matrix Invertibility Using Determinants

A matrix is invertible if its determinant is non-zero. Therefore, to find the real values of \(c\) that make each matrix invertible, we need to evaluate the determinant and find for which values of \(c\), the determinant is not equal to zero.

Step 2a: Calculate Determinant for Matrix (a)

The matrix is \(\begin{bmatrix}1 & 0 & 3 \3 & -4 & c \2 & 5 & 8\end{bmatrix}\). The determinant is calculated by:\[ 1(-4*8 - c*5) - 0(3*8 - c*2) + 3(3*5 + 4*2) \]Simplify to find:\[ det(A) = -32 - 5c + 39 + 12 = 19 - 5c\]Set \(19 - 5c eq 0\) to determine invertibility: \[ c eq \frac{19}{5} \]

Step 2b: Calculate Determinant for Matrix (b)

The matrix is \(\begin{bmatrix}0 & c & -c \-1 & 2 & 1 \c & -c & c\end{bmatrix}\).Calculate the determinant as follows:\[0(2c - 1(-c)) - c((-1)c - c(1)) + (-c)((-1)(-c) - 2c) \]Simplify to:\[det(B) = 0 - c(-c - c) - c(c + 2c) = 2c^2 + 3c^2 = 5c^2\]Set \(5c^2 eq 0\):\[ c eq 0 \]

Step 2c: Calculate Determinant for Matrix (c)

The matrix is \(\begin{bmatrix}c & 1 & 0 \0 & 2 & c \-1 & c & 5\end{bmatrix}\).The determinant is:\[c(2*5 - c*1) - 1(0(5) - c(-1)) \]Simplify to:\[det(C) = 10c - c^2 + c^2 = 10c\]Set \(10c eq 0\):\[ c eq 0 \]

Step 2d: Calculate Determinant for Matrix (d)

The matrix is \(\begin{bmatrix}4 & c & 3 \c & 2 & c \5 & c & 4\end{bmatrix}\).The determinant is:\[det(A) = 4(2*4 - c^2) - c(c*4 - 3*4) + 3(c^2 - 10)\]Simplify to:\[32 - 4c^2 + 3c - 12 + 3c^2 - 30 \]\[det(D) = -c^2 - 37 \]Set \(-c^2 - 37 eq 0\): Since \(c^2 + 37\) can never be zero (a sum of positive numbers), any real \(c\) makes the matrix invertible.

Step 2e: Calculate Determinant for Matrix (e)

The matrix is \(\begin{bmatrix}1 & 2 & -1 \0 & -1 & c \2 & c & 1\end{bmatrix}\).The determinant is:\[1(-1*1 - c*2) - 2(0*1 - c*2) - 1(0 - c*2)\]Simplify to:\[det(E) = -1 - 2c + 4c - 2c = -1\]Since the determinant is always -1 regardless of \(c\), the matrix is always invertible.

Step 2f: Calculate Determinant for Matrix (f)

The matrix is \(\begin{bmatrix}1 & c & -1 \c & 1 & 1 \0 & 1 & c\end{bmatrix}\).The determinant is:\[1(1*c - 1*1) - c(c*c - c*0) - 1 (1*1 - c*0) \]Simplify to:\[det(F) = c - 1 - c^2 - 1\]Rearrange:\[-c^2 + c - 2\]Set \(-c^2 + c - 2 eq 0\). Solve this quadratic:\[ c^2 - c + 2 = 0 \] has no real solutions, therefore all real \(c\) make the matrix invertible.

Key concepts

Determinants

Determinants are central in matrix algebra and are critical for analyzing the invertibility of matrices. The determinant of a matrix is a special number that can be calculated from its elements. For a 3x3 matrix, the determinant helps determine whether a matrix is invertible. To find the determinant of a 3x3 matrix, apply the formula:\[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]where \(a, b, c, d, e, f, g, h,\) and \(i\) are elements of the matrix. Calculating determinants can seem tricky at first, but with practice, the steps become clearer. It's a systematic method of expanding along one row or column, considering sub-matrices formed by excluding the current row and column.A key property of determinants is that if the determinant of a matrix is zero, the matrix is not invertible. Conversely, a non-zero determinant indicates that the matrix can be inverted, which is crucial in solving systems of linear equations.

Real Values of c

When analyzing matrix invertibility in the context of variable matrices, understanding the real values of \(c\) that affect the determinant is essential. For each matrix given in the problem, we compute the determinant and solve for \(c\) such that the determinant is not zero.

• For Matrix (a): The determinant is calculated as \(19 - 5c\). To ensure invertibility, set \(19 - 5c eq 0\) yielding \(c eq \frac{19}{5}\).
• For Matrix (b): The determinant is \(5c^2\). Thus, \(c eq 0\) since any non-zero value ensures invertibility.
• For Matrix (c): The determinant becomes \(10c\). Hence, \(c eq 0\) for invertibility.

Such analysis is crucial when dealing with parameterized matrices, offering insights into which specific conditions (values of \(c\)) will affect a matrix's ability to be inverted.

Matrix Algebra

Matrix algebra is a foundational mathematical language used to handle data, solve systems of equations, and perform numerous calculations. It involves operations such as addition, subtraction, multiplication, and, importantly, finding the inverse of matrices.One fundamental aspect of matrix algebra is determining if a matrix can be inverted, also known as finding the inverse matrix. This involves calculating the determinant and ensuring it is non-zero. In applications ranging from physics to computer science, matrix algebra helps tackle complex problems efficiently.Understanding how to manipulate matrices, including finding inverses, is vital for solving linear equations. For instance, given a system of equations represented in matrix form \(AX = B\), using the inverse of \(A\) (if it exists) allows you to find \(X = A^{-1}B\). This technique is powerful and widely used in computational solutions.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vector spaces and linear mappings between these spaces. It is rich in geometric and algebraic structures that allow us to solve many practical problems in science and engineering. Key components of linear algebra include vectors, matrices, and operations that link them, such as dot products and cross products. Determinants and their calculation play an essential role in linear algebra for understanding the properties of linear transformations represented by matrices. Linear algebra is not just theoretical; it underpins computer graphics, data science, machine learning, and more. Being able to determine matrix invertibility is a vital skill because it relates to solving linear systems, assessing the stability of mathematical models, and performing eigenvalue and eigenvector analyses. Overall, linear algebra provides the tools to navigate and transform data in various dimensions, making it an indispensable field of study in both pure and applied mathematics.