Question

If the trace of the matrix \(\mathrm{A}\) is 5, and the trace of the matrix \(\mathrm{B}\) is 7, then find trace of the matrix \((3 \mathrm{~A}+\) \(2 \mathrm{~B}\) ). (1) 12 (2) 29 (3) 19 (4) None of these

Short answer

Answer: The trace of the matrix (3A + 2B) is 29.

Step-by-step solution

Recall the definition of trace

The trace of a square matrix is the sum of the elements on its main diagonal (from top-left to bottom-right). Mathematically, it is denoted by Tr(A) and is given by: Tr(A) = \(\sum_{i=1}^n a_{ii}\), where A is an n×n matrix and \(a_{ii}\) are its diagonal elements.

Recall the properties of trace operation

The trace operation has the following properties: 1. Tr(kA) = kTr(A), where k is a scalar constant and A is a square matrix. 2. Tr(A + B) = Tr(A) + Tr(B), where A and B are square matrices of the same order.

Apply properties to find the trace of the given matrix combination

We are given that Tr(A) = 5 and Tr(B) = 7, and we want to find the trace of the matrix (3A + 2B). Using the properties of the trace operation, we can rewrite this as: Tr(3A + 2B) = Tr(3A) + Tr(2B) Now, apply property 1 from above to both terms on the right side: Tr(3A + 2B) = 3Tr(A) + 2Tr(B) We know the values of Tr(A) and Tr(B) from the problem statement, so substitute them in: Tr(3A + 2B) = 3(5) + 2(7)

Calculate the trace value

Now, calculate the value of the expression by performing the operations: Tr(3A + 2B)= 15 + 14 = 29 So, the trace of the matrix (3A + 2B) is 29. Therefore, the correct answer is option (2) 29.

Key concepts

Matrix Properties

Matrices are fascinating mathematical objects used extensively in various fields. They have several properties that are crucial for understanding operations involving matrices. One such property is the **trace** of a matrix.

• **Trace**: The trace of a matrix is the sum of its diagonal elements. It is often denoted by Tr(A) for a matrix A.
• **Square matrices**: The trace is only defined for square matrices, meaning the number of rows and columns must be equal.

One essential property of traces is linearity. This means:

• If you add two matrices, A and B, the trace of their sum is the sum of their traces: \( Tr(A + B) = Tr(A) + Tr(B) \).
• If you multiply a matrix A by a scalar k, then the trace of the result is k times the trace of the original matrix: \( Tr(kA) = k \times Tr(A) \).

These properties help simplify complex expressions involving multiple matrices. By leveraging these properties, we can simplify the process of finding the trace of combined matrices.

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a constant value, known as a scalar. This operation is straightforward and changes the size of the matrix elements proportionally.
To perform scalar multiplication on a matrix:

• Identify the scalar, say k, which you will use to multiply the matrix.
• Multiply each element of the matrix by that scalar.

Consider a matrix A and a scalar k. The resulting matrix after multiplication will have each element modified as:
\[ (kA)_{ij} = k \times a_{ij} \]
where \(a_{ij}\) is the element from matrix A.
This operation is vital in linear algebra and can alter various properties of the matrix, including the trace. For instance, if Tr(A) is the trace of A, then the trace of kA is k times Tr(A), allowing you to quickly determine the trace without recalculating from scratch.

Matrix Addition

Matrix addition is the operation where elements of two matrices are combined into a new matrix by adding corresponding elements. It's fundamental in many areas of math and science.

• To add matrices A and B, they must be of the same size, meaning the same number of rows and columns.
• The resulting matrix, matrix C, will have elements calculated as: \[ c_{ij} = a_{ij} + b_{ij} \]

This operation supports the trace property: the trace of a matrix sum is the sum of their traces. Therefore, \( Tr(A + B) = Tr(A) + Tr(B) \), which is used extensively to simplify calculations involving linear combinations of matrices.