Question

Determine whether the statement is true or false. Justify your answer. $$ \left[\begin{array}{rr} 3 & 2 \\ 1 & -4 \end{array}\right]\left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{rr} 3 & 2 \\ 1 & -4 \end{array}\right] $$

Short answer

The statement is true. Both sides of the equation result in the same matrix, i.e., \(\left[\begin{array}{rr}3 & 2 \1 & -4 \end{array}\right]\).

Step-by-step solution

Multiply the Matrices on the Left

Let's multiply the first two matrices on the left. The matrix product is calculated as \[\left[\begin{array}{rr}3 & 2 \1 & -4 \end{array}\right]\left[\begin{array}{rr}1 & 0 \0 & 1 \end{array}\right]=\left[\begin{array}{rr}3*1+2*0 & 3*0+2*1 \1*1-4*0 & 1*0 - 4*1 \end{array}\right]=\left[\begin{array}{rr}3 & 2 \1 & -4 \end{array}\right].\]

Multiply the Matrices on the Right

Now, we calculate the product of the last two matrices on the right.\[\left[\begin{array}{rr}1 & 0 \0 & 1 \end{array}\right]\left[\begin{array}{rr}3 & 2 \1 & -4 \end{array}\right]=\left[\begin{array}{rr}1*3+0*1 & 1*2+0*-4 \0*3+1*1 & 0*2 + 1*-4 \end{array}\right]=\left[\begin{array}{rr}3 & 2 \1 & -4 \end{array}\right].\]

Compare the Results

After performing the matrix multiplication, it is evident that the result of both multiplications is identical. Therefore, the statement \(\left[\begin{array}{rr}3 & 2 \1 & -4 \end{array}\right]\left[\begin{array}{rr}1 & 0 \0 & 1 \end{array}\right]=\left[\begin{array}{ll}1 & 0 \0 & 1 \end{array}\right]\left[\begin{array}{rr}3 & 2 \1 & -4 \end{array}\right]\) is true.

Key concepts

Linear Algebra

Linear Algebra is a branch of mathematics that focuses on the study of vectors, vector spaces (also known as linear spaces), linear transformations, and systems of linear equations. It provides a framework for understanding and solving problems involving lines, planes, and higher-dimensional analogs, fundamentally related to coordinate and vector space theories.

One of its primary tools is matrix multiplication, a process that combines two matrices to produce a third matrix that represents the composite transformation. In applying linear algebra to real-world problems, matrices are used to represent data or connections between various quantities, making it an essential concept in areas ranging from computer science to engineering and economics.

Identity Matrix

The Identity Matrix, often represented as an uppercase 'I', plays a vital role in matrix operations. It's analogous to the number 1 in arithmetic. Just as multiplying any number by 1 gives the original number, multiplying any matrix by the identity matrix of appropriate size will return the original matrix.

The identity matrix is a square matrix with '1's on the main diagonal (from the top-left to the bottom-right corners) and '0's everywhere else. The property illustrated in the original textbook exercise where the identity matrix maintains the original matrix upon multiplication is crucial for demonstrating that certain matrices are invertible, and it is a foundational element in the definition of matrix inverses.

Matrix Properties

Matrices possess a range of properties that govern operations such as addition, subtraction, and multiplication. For instance, matrix multiplication is associative, meaning that when you have three or more matrices, the way in which they are grouped during multiplication does not affect the final product. Additionally, matrix multiplication is distributive over matrix addition.

However, unlike regular multiplication, matrix multiplication is not always commutative; meaning the order in which matrices are multiplied can affect the outcome. The exercise in question illustrates a special case where two matrices can be multiplied in either order, thanks to one being an identity matrix, without altering the result—a scenario which doesn't often occur with different non-identity matrices.

Algebraic Structures

Algebraic structures are mathematical frameworks characterized by a set of elements and one or more operations that follow specific rules or axioms. These structures include groups, rings, fields, and vector spaces, among others.

A key example in linear algebra is a vector space, which is an algebraic structure formed by vectors, requiring both a field and vector addition and scalar multiplication operations to fulfill certain axioms. This structure allows us to generalize and work with concepts of linearity across various dimensions and applications. The matrices themselves, under certain operations, can form algebraic structures, such as a ring under matrix addition and multiplication, which can further aid in solving linear equations and transformations.