Determinants and Cramer's Rule: Videos & Practice Problems

In the study of matrices, one important concept is the determinant, which is a scalar value that can be calculated from a square matrix. Understanding how to compute the determinant is essential, especially when solving systems of equations. The determinant provides insights into the properties of the matrix, such as whether it is invertible.

For a 2x2 matrix, the determinant can be calculated using a straightforward formula. Given a matrix represented as:

\[\begin{pmatrix}a & b \\c & d\end{pmatrix}\]

the determinant is calculated using the formula:

\[\text{det}(A) = ad - bc\]

Here, \(a\) and \(d\) are the elements on the main diagonal (from the top left to the bottom right), while \(b\) and \(c\) are the elements on the other diagonal (from the top right to the bottom left). The process involves multiplying the elements of each diagonal and then subtracting the product of the second diagonal from the product of the first.

For example, consider the matrix:

\[\begin{pmatrix}3 & 2 \\5 & 4\end{pmatrix}\]

To find the determinant, we calculate:

\[\text{det}(A) = (3 \times 4) - (2 \times 5) = 12 - 10 = 2\]

This means the determinant of this matrix is 2. This process can be applied to any 2x2 matrix, allowing for the evaluation of its determinant easily.

Let’s look at another example with the matrix:

\[\begin{pmatrix}8 & 4 \\5 & 0\end{pmatrix}\]

Using the determinant formula, we find:

\[\text{det}(A) = (8 \times 0) - (4 \times 5) = 0 - 20 = -20\]

Thus, the determinant of this matrix is -20. It’s important to note that the determinant can also be calculated with negative numbers. For instance, with the matrix:

\[\begin{pmatrix}-3 & -7 \\-2 & 1\end{pmatrix}\]

the calculation would be:

\[\text{det}(B) = (-3 \times 1) - (-7 \times -2) = -3 - 14 = -17\]

In summary, the determinant is a crucial concept in linear algebra, providing valuable information about the matrix's characteristics. Mastering the calculation of determinants for 2x2 matrices lays the groundwork for more complex matrix operations and applications in various mathematical fields.

Cramer's Rule is a method used to solve a system of linear equations with two variables by utilizing determinants of matrices. To apply Cramer's Rule, you first need to express the system of equations in matrix form. For example, consider the equations:

1. \(2x + y = 5\)

2. \(-4x + 6y = -2\)

These can be represented in an augmented matrix format, where the coefficients of \(x\) and \(y\) are organized into a matrix alongside the constants:

\[\begin{bmatrix}2 & 1 & | & 5 \\-4 & 6 & | & -2\end{bmatrix}\]

In Cramer's Rule, the solution for \(x\) and \(y\) is found using the determinants of matrices. The formula for \(x\) is given by:

\[x = \frac{D_x}{D}\]

And for \(y\):

\[y = \frac{D_y}{D}\]

Where \(D\) is the determinant of the coefficient matrix:

\[D = \begin{vmatrix}2 & 1 \\-4 & 6\end{vmatrix} = (2)(6) - (1)(-4) = 12 + 4 = 16\]

Next, to find \(D_x\), replace the first column of the coefficient matrix with the constants from the right side of the equations:

\[D_x = \begin{vmatrix}5 & 1 \\-2 & 6\end{vmatrix} = (5)(6) - (1)(-2) = 30 + 2 = 32\]

For \(D_y\), replace the second column with the constants:

\[D_y = \begin{vmatrix}2 & 5 \\-4 & -2\end{vmatrix} = (2)(-2) - (5)(-4) = -4 + 20 = 16\]

Now, substituting these determinants back into the formulas gives:

\[x = \frac{D_x}{D} = \frac{32}{16} = 2\]

\[y = \frac{D_y}{D} = \frac{16}{16} = 1\]

Thus, the solution to the system of equations is \(x = 2\) and \(y = 1\). You can verify this solution by substituting these values back into the original equations to ensure both equations hold true. Cramer's Rule is a powerful tool for solving linear systems, especially when dealing with larger matrices, as it provides a systematic approach to finding solutions through determinants.

To calculate the determinant of a 3 by 3 matrix, the process is slightly more complex than for a 2 by 2 matrix, but it can be broken down into manageable steps involving 2 by 2 determinants. The determinant of a 3 by 3 matrix can be expressed using the formula:

For a matrix A represented as:

\[A = \begin{pmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{pmatrix}\]

the determinant is calculated as:

\[\text{det}(A) = a_{11} \cdot \text{det}(M_{11}) - a_{12} \cdot \text{det}(M_{12}) + a_{13} \cdot \text{det}(M_{13})\]

where M represents the minor matrices obtained by removing the row and column of the respective element.

To illustrate, consider a 3 by 3 matrix:

\[\begin{pmatrix}3 & 1 & 0 \\2 & -3 & -1 \\4 & 6 & 0\end{pmatrix}\]

To find the determinant, we first identify the elements in the first row: 3, 1, and 0. The signs alternate starting with a plus for the first term:

• For the first term, take 3 and calculate the determinant of the minor matrix:

\[\begin{pmatrix}-3 & -1 \\6 & 0\end{pmatrix}\]

which gives:

\[\text{det} = (-3)(0) - (-1)(6) = 6\]
• For the second term, take 1 and calculate the determinant of the minor matrix:

\[\begin{pmatrix}2 & -1 \\4 & 0\end{pmatrix}\]

which gives:

\[\text{det} = (2)(0) - (-1)(4) = 4\]
• For the third term, take 0, which will result in zero regardless of the minor matrix.

Putting it all together, we have:

\[\text{det}(A) = 3 \cdot 6 - 1 \cdot 4 + 0 \cdot \text{(any value)} = 18 - 4 + 0 = 14\]

Thus, the determinant of the 3 by 3 matrix is 14. This method effectively reduces the complexity of calculating a 3 by 3 determinant by leveraging the simpler 2 by 2 determinants, making the process more manageable.

To solve a system of three equations with three unknowns using Cramer's rule, we first need to understand the structure of the equations and how to represent them in matrix form. Cramer's rule provides a straightforward method to find the values of the unknowns by using determinants of matrices.

Given a system of equations, we can express it in the form of an augmented matrix. For example, if we have the equations:

\[\begin{align*}-5x - y + 4z &= 4 \\0x + 3y + 6z &= 21 \\x + y + z &= 6\end{align*}\]

We can create the augmented matrix by organizing the coefficients of the variables and the constants:

\[\begin{bmatrix}-5 & -1 & 4 & | & 4 \\0 & 3 & 6 & | & 21 \\1 & 1 & 1 & | & 6\end{bmatrix}\end{p}

In Cramer's rule, we define a determinant \(D\) (denoted as \(D\)) which is the determinant of the coefficient matrix formed by the coefficients of \(x\), \(y\), and \(z\). If \(D\) is zero, the system has no unique solution. If \(D\) is non-zero, we can find the solutions for \(x\), \(y\), and \(z\) using the following formulas:

\[x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}\end{p}

Where \(D_x\), \(D_y\), and \(D_z\) are the determinants of matrices formed by replacing the respective columns of the coefficient matrix with the constants from the right side of the equations.

To calculate \(D\), we use the formula for the determinant of a 3x3 matrix:

\[D = a(ei - fh) - b(di - fg) + c(dh - eg)\end{p}

Where the matrix is represented as:

\[\begin{bmatrix}a & b & c \\d & e & f \\g & h & i\end{bmatrix}\end{p}

After calculating \(D\), we proceed to find \(D_x\), \(D_y\), and \(D_z\) by substituting the appropriate columns with the constants. For instance, to find \(D_x\), we replace the first column (the coefficients of \(x\)) with the constants:

\[D_x = \begin{vmatrix}4 & -1 & 4 \\21 & 3 & 6 \\6 & 1 & 1\end{vmatrix}\end{p}

Similarly, we calculate \(D_y\) and \(D_z\) by replacing the second and third columns respectively. Once we have all determinants, we can substitute them back into the formulas for \(x\), \(y\), and \(z\) to find their values.

For example, if we find \(D = -3\), \(D_x = -15\), \(D_y = 15\), and \(D_z = -18\), we can compute:

\[x = \frac{-15}{-3} = 5, \quad y = \frac{15}{-3} = -5, \quad z = \frac{-18}{-3} = 6\end{p}

Thus, the solution to the system of equations is \(x = 5\), \(y = -5\), and \(z = 6\).