Question

If \(\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\left(\begin{array}{cc}a & b+c \\ b-c & d\end{array}\right)=\left(\begin{array}{rr}4 & -5 \\ 3 & 2\end{array}\right)\), then \((a-b)+(c-d)=\) (1) \(-2\) (2) 9 (3) 2 (4) \(-1\)

Short answer

In this problem, we are given the product of two matrices which results in a third matrix. We are asked to find the values of a, b, c, and d, and then evaluate the expression (a-b)+(c-d). After performing the matrix multiplication and solving for the unknown values, we find that a = 4, b = -1, c = -4, and d = 2. Evaluating the expression gives us (a-b)+(c-d) = (-1), which is the answer.

Step-by-step solution

Matrix Multiplication

To begin, let's recall the matrix multiplication rule. When multiplying matrices, we multiply each element of the first matrix's row with the corresponding element of the second matrix's column and sum them together to get the resulting matrix's elements. In this case, the matrices are 2x2, so the matrix multiplication rule can be written as: $ \begin{array}{cc} 1*a+0*(b-c) & 1*(b+c)+0*d \\ 0*a+1*(b-c) & 0*(b+c)+1*d \end{array} $ This will result in the given product matrix: $ \begin{array}{cc} 4 & -5 \\ 3 & 2 \end{array} $

Find the values of a, b, c, and d based on the multiplication result

Now, let's find the values of a, b, c, and d. Comparing the elements of the matrices, we can set up the following equations: 1. \(a = 4\) 2. \(b + c = -5\) 3. \(b - c = 3\) 4. \(d = 2\)

Solve the equations for the unknowns

We already have the values for a and d. We now need to solve equations 2 and 3 to find the values of b and c. Equation 3 directly gives us the value of b: \(b = 3+c\). We can substitute this expression for b into equation 2: \((3+c) + c = -5\) Solve for c: \(c = -4\) Now use the value of c to find the value of b: \(b = 3 + (-4) = -1\) So, the values of a, b, c, and d are: \(a = 4\) \(b = -1\) \(c = -4\) \(d = 2\)

Calculate the expression (a-b)+(c-d)

Now that we have the values of a, b, c, and d, we can compute the expression: (a-b)+(c-d)=\(\operatorname{(4-(-1))+(-4-2)}\) \(=(4+1)+(-4-2)\) \(=5+-6\) \(=-1\)

Final answer

Based on our calculations, the value of (a-b)+(c-d)=-1, which corresponds to option (4).

Key concepts

Linear Algebra

Understanding linear algebra is crucial for solving various types of mathematical and real-world problems, particularly when dealing with multiple dimensions. Linear algebra is a field of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. Matrices, which are rectangular arrays of numbers, are an essential tool in this discipline.

Matrices can represent systems of linear equations, transformations in space, and more. Each element in the matrix can be considered a coordinate in a higher-dimensional space. Operations like matrix multiplication, which we see in the given exercise, are fundamental in linear algebra as they allow us to combine linear transformations or, as in our case, to compare coefficients from different sets of simultaneous equations to derive values of unknown variables.

Systems of Equations

In linear algebra, systems of equations are sets of equations where the variables appear in an interconnected fashion. Solving them involves finding the values for the variables that satisfy all equations simultaneously. In our exercise, we are presented with a particular instance where the system is represented by matrix multiplication involving 2x2 matrices.

To approach a system of equations, especially when derived from matrix multiplication, you sequentially solve for one variable and then substitute this value into the other equations, a process that can be seen in steps 2 and 3. It's a systematic method of elimination that requires careful manipulation and substitution until all variables are found. The result provides insight into how different variables interact within a given system, predicted by the structures of the original matrices.

2x2 Matrices

A 2x2 matrix is a square matrix that has two rows and two columns. In our exercise, we're dealing with 2x2 matrices during the matrix multiplication process. Given its simplicity, it is a common illustration for introducing fundamental concepts in linear algebra such as matrix operations, determinants, and inverses.

A 2x2 matrix can encapsulate a transformation in a two-dimensional space or represent a system of two linear equations. When multiplying two 2x2 matrices, each element of the resulting matrix is the sum of the products of elements in the corresponding row of the first matrix and the column of the second matrix. The operation has real-world implications, such as transforming geometric figures and solving coupled linear equations — a kind of problem frequently encountered in physics and engineering.