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Chapter 3: Problem 27

Find conditions on \(a, b, c,\) and \(d\) such that \(B=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) commutes with every \(2 \times 2\) matrix.

Short Answer

Expert verified
Matrix \(B\) must be a scalar multiple of the identity matrix: \(B = aI\).

Step by step solution

01

Define Commutativity

For any matrix \(B\) to commute with every \(2 \times 2\) matrix, \(B\) must satisfy \(AB = BA\) for all \(2 \times 2\) matrices \(A\). A specific case to consider is when \(A = I\), the identity matrix. But the condition must hold for any \(A\), not just specific ones.
02

Matrix Multiplication

Let \(A = \left[\begin{array}{cc} x & y \ z & w \end{array}\right]\). Compute \(AB\) and \(BA\) using matrix multiplication. Calculate \(AB = \begin{bmatrix} xa + yc & xb + yd \ za + wc & zb + wd \end{bmatrix}\) and \(BA = \begin{bmatrix} ax + bz & ay + bw \ cx + dz & cy + dw \end{bmatrix}\).
03

Equate Matrices

For \(AB = BA\) to hold for all \(x, y, z, w\), equate corresponding components of \(AB\) and \(BA\). We derive the following equations: 1. \(xa + yc = ax + bz\) 2. \(xb + yd = ay + bw\) 3. \(za + wc = cx + dz\) 4. \(zb + wd = cy + dw\)
04

Analyze Each Equation

Since equations 1 and 2 must hold for all values of \(x, y, z, w\), analyze them for individual terms:- Compare terms: \(xa = ax\), \(yc = bz\), \(xb = ay\), and \(yd = bw\). This constrains \(a = d\), \(b = c\), and leads to further implications.
05

Derive Conditions

From the constraints, deduce the conditions:- For \(yc = bz\), since \(y\) and \(z\) are independent, and \(c = b\) previously derived, let \(yc = 0\). Thus, \(b = 0\) for it to trivially hold.- Similarly, from earlier steps, we require \(a = d\) and both \(b = 0\) and \(c = 0\). Hence \(B = \begin{bmatrix} a & 0 \ 0 & a \end{bmatrix} = aI\), where \(I\) is the identity matrix.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra, where two matrices are combined to form another matrix. This operation is not the same as multiplying individual numbers. Instead, it involves a series of additions and multiplications across rows and columns of the matrices involved.

To multiply two matrices, one must follow these steps:
  • Ensure the number of columns in the first matrix matches the number of rows in the second matrix.
  • Take the entries across a row of the first matrix and multiply each with the corresponding entry down a column in the second matrix.
  • Sum the results of these multiplications to form an entry in the resulting matrix.
For example, when you multiply a 2x2 matrix A with another 2x2 matrix B, the resulting matrix will also be 2x2. The formula for each element in the resulting matrix is derived from these row-by-column multiplications. This operation lays the groundwork for more complex concepts like matrix commutativity.
Identity Matrix
An identity matrix is like the number 1 in matrix algebra, in that it doesn鈥檛 change other matrices when used in multiplication. For any square matrix of size n, the identity matrix I is an n-by-n matrix with ones on the diagonal and zeroes elsewhere.

Key properties of the identity matrix include:
  • For any matrix A of the same size, multiplying by the identity matrix (i.e., AI or IA) results in A itself.
  • It effectively acts as a neutral element in matrix multiplication.
The presence of the identity matrix is crucial in finding the inverse of matrices and understanding matrix operations and transformations in geometric space. In the context of the problem, checking commutativity of matrix B with the identity matrix helps simplify understanding what forms matrix B would take to commute with any matrix.
Conditions for Commutativity
Commutativity in the context of matrices means that for two matrices A and B, the multiplication AB yields the same result as BA. However, unlike with real numbers, commutativity is not generally a given with matrices, adding complexity to linear algebra.

For a matrix B to commute with any other 2x2 matrix, including a generic identity matrix, it must take on a specific form. The rigorous analysis of matrix multiplication shows that:
  • Each corresponding element in the product matrices AB and BA must be equal.
  • This equality imposes conditions on the individual elements of matrix B itself.
  • In this exercise, these conditions lead to the matrix B needing to be a scalar multiple of the identity matrix: B = \(aI\).
Hence, the conditions derived ensure that when matrix B is multiplied by any other 2x2 matrix, the result will consistently match no matter the order of multiplication, fulfilling the commutativity requirement.

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Most popular questions from this chapter

Suppose the Leslie matrix for the VW beetle is \(L=\) \(\left[\begin{array}{lll}0 & 0 & 20 \\ 0.1 & 0 & 0 \\ 0 & 0.5 & 0\end{array}\right] .\) Starting with an arbitrary \(\mathbf{x}_{0},\) deter- mine the behavior of this population.

Prove that, for \(m \times n\) matrices \(A\) and \(B\), rank \((A+B) \leq\) \(\operatorname{rank}(A)+\operatorname{rank}(B)\).

Prove that \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is a linear transformation if and only if \\[ T\left(c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}\right)=c_{1} T\left(\mathbf{v}_{1}\right)+c_{2} T\left(\mathbf{v}_{2}\right) \\] for all \(\mathbf{v}_{1}, \mathbf{v}_{2}\) in \(\mathbb{R}^{n}\) and scalars \(c_{1}, c_{2}\).

Find the standard matrix of the composite transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\) ? Clockwise rotation through \(45^{\circ}\), followed by projection onto the \(y\) -axis, followed by clockwise rotation through \(45^{\circ}\)

(a) Prove that a graph is bipartite if and only if its vertices can be labeled so that its adjacency matrix can be partitioned as \\[ A=\left[\begin{array}{ll} O & B \\ B^{T} & O \end{array}\right] \\] (b) Using the result in part (a), prove that a bipartite graph has no circuits of odd length.
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