Problem 27 - Precalculus: Functions and Graphs

Question

Verify the Identity by expanding each determinant. $$\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|+\left|\begin{array}{ll} a & e \\ c & f \end{array}\right|=\left|\begin{array}{ll} a & b+e \\ c & d+f \end{array}\right|$$

Step-by-Step Solution

Answer

The identity is verified: both sides expand to \(ad + af - bc - ce\).

Step 1: Expand the First Determinant

Start by expanding the first determinant \(\left|\begin{array}{ll} a & b \ c & d \end{array}\right|\). The expansion formula for a 2x2 determinant is \(ad - bc\). Here, \(a\), \(b\), \(c\), and \(d\) are the elements of the determinant. This expansion gives \(ad - bc\).

Step 2: Expand the Second Determinant

Next, expand the second determinant \(\left|\begin{array}{ll} a & e \ c & f \end{array}\right|\) using the same formula \(ad - bc\). Substitute \(a\), \(e\), \(c\), and \(f\) respectively into the formula, yielding \(af - ce\).

Step 3: Combine the Expanded Determinants

Now, add the results of the expansion from Step 1 and Step 2. Therefore, the expression becomes \((ad - bc) + (af - ce)\). When simplified, this results in \(ad - bc + af - ce\).

Step 4: Expand the Combined Determinant on Right Side

Next, expand the determinant on the right side, \(\left|\begin{array}{ll} a & b+e \ c & d+f \end{array}\right|\). Using the expansion formula \(ad - bc\), plug in the expressions to get \(a(d+f) - c(b+e)\). Simplify this to \(ad + af - cb - ce\).

Step 5: Verify Equality

Lastly, compare the results from Step 3 and Step 4. Both expressions \(ad + af - bc - ce\) and \(ad + af - bc - ce\) are identical. Therefore, the identity is verified.

Key Concepts

2x2 Determinant

A 2x2 determinant is a mathematical calculation that provides a single number representing certain properties of a 2x2 matrix. A matrix in its 2x2 form is a square with two rows and two columns. It looks like this: \[ \begin{array}{ll} a & b \ c & d \end{array} \] The determinant for this kind of matrix is calculated using the formula: \[ ad - bc \] where \(a\), \(b\), \(c\), and \(d\) are elements of the matrix.

• Multiply \(a\) by \(d\)
• Multiply \(b\) by \(c\)
• Subtract the second product from the first

The determinant gives us insights into important characteristics such as matrix invertibility. If the determinant is zero, the matrix is non-invertible, meaning it does not have an inverse matrix. Otherwise, it is invertible.

Matrix Algebra

Matrix algebra involves operations like addition, subtraction, and multiplication of matrices. When verifying determinants, addition of two matrices or operations involving their determinants can be crucial. Consider two 2x2 matrices: \[ \begin{array}{ll} a & b \ c & d \end{array} \] and \[ \begin{array}{ll} a & e \ c & f \end{array} \] In matrix algebra, you can find the determinants of these matrices separately, calculate each one using the determinant formula, and combine their values according to the problem's requirements, like our exercise does. Moreover, matrices in algebra follow specific rules when performing operations:

• Addition: Add corresponding elements from each matrix.
• Subtraction: Subtract corresponding elements from each matrix.
• Multiplication (not covered in the exercise, but useful to know): It's different than standard number multiplication and involves the dot product of rows and columns.

These operations are fundamental for expressing and solving more complex algebraic problems involving matrices.

Identity Verification

Identity verification in matrix problems involves showing that two expressions are mathematically equivalent. This process often includes multiple steps of expanding and simplifying matrices. In our exercise, the identity we're verifying involves ensuring that the sum of two separate 2x2 determinants is equivalent to the determinant of a combined matrix. To do this, you follow these steps:

• Expand each determinant separately using the formula \(ad - bc\) for 2x2 matrices.
• Combine the results to form a summed expression.
• Expand the determinant of the given matrix on the right side of the equation.
• Verify that both sides are equal by simplifying each to show they have the same terms.

When both expressions on each side of an equation match after these steps, the identity is verified. This shows the math principles in action, effectively illustrating the relationships between matrix components in equal terms.