Problem 39 - Advanced Engineering Mathematics

Question

Suppose \(\mathbf{A}\) and \(\mathbf{B}\) are \(n \times n\) matrices and \(\mathbf{A}\) is nonsingular. Show that if \(\mathbf{A} \mathbf{B}=\mathbf{0}\), then \(\mathbf{B}=\mathbf{0}\).

Step-by-Step Solution

Answer

If \(\mathbf{A}\mathbf{B} = \mathbf{0}\) and \(\mathbf{A}\) is nonsingular, then \(\mathbf{B} = \mathbf{0}\).

Step 1: Understand the Problem

We are given two \(n \times n\) matrices \(\mathbf{A}\) and \(\mathbf{B}\), and it is known that \(\mathbf{A}\) is nonsingular, meaning \(\mathbf{A}\) has an inverse. We aim to show that if the product of \(\mathbf{A}\) and \(\mathbf{B}\) is the zero matrix, then \(\mathbf{B}\) itself must be the zero matrix.

Step 2: Utilize Matrix Inverse

Since \(\mathbf{A}\) is nonsingular, it has an inverse denoted by \(\mathbf{A}^{-1}\). We will use this inverse to manipulate the equation \(\mathbf{A}\mathbf{B} = \mathbf{0}\) to solve for \(\mathbf{B}\).

Step 3: Multiply by the Inverse

Multiply both sides of the equation \(\mathbf{A}\mathbf{B} = \mathbf{0}\) by \(\mathbf{A}^{-1}\) from the left:\[\mathbf{A}^{-1}(\mathbf{A}\mathbf{B}) = \mathbf{A}^{-1}\mathbf{0}.\]

Step 4: Simplify Using Matrix Properties

Applying the property of the inverse, we know \(\mathbf{A}^{-1}\mathbf{A} = \mathbf{I}\), where \(\mathbf{I}\) is the identity matrix. Hence, we have:\[\mathbf{I}\mathbf{B} = \mathbf{0}.\]

Step 5: Conclude the Proof

Since \(\mathbf{I}\mathbf{B} = \mathbf{B}\) for any matrix \(\mathbf{B}\), the equation \(\mathbf{I}\mathbf{B} = \mathbf{0}\) simplifies to \(\mathbf{B} = \mathbf{0}\). Thus, we have shown that \(\mathbf{B}\) must be the zero matrix.

Key Concepts

Nonsingular Matrix

A nonsingular matrix, often referred to as an invertible or non-degenerate matrix, is one with a very important property: it has an inverse. In simpler terms, if you have a matrix \( \mathbf{A} \), it's considered nonsingular if you can find another matrix \( \mathbf{A}^{-1} \) such that multiplying them yields the identity matrix \( \mathbf{I} \).

• Mathematically, this can be represented as \( \mathbf{A} \mathbf{A}^{-1} = \mathbf{A}^{-1} \mathbf{A} = \mathbf{I} \).
• The identity matrix \( \mathbf{I} \) is like multiplying a number by 1; it leaves the original matrix unchanged.

For a matrix to be nonsingular, it must be square (having the same number of rows and columns) and its determinant must not be zero. Nonsingular matrices are crucial in linear algebra because they ensure certain linear transformations are reversible, meaning they can "undo" operations.

Inverse of a Matrix

The inverse of a matrix is like the matrix equivalent of a reciprocal of a number. If you think about how multiplying a number by its reciprocal gives you 1, a matrix multiplied by its inverse gives you the identity matrix.

• The notation \( \mathbf{A}^{-1} \) is used to represent the inverse of \( \mathbf{A} \).
• Only nonsingular matrices have inverses, which is a key point to remember.

To find the inverse, various methods can be used, including row reduction or utilizing determinants and adjugates for smaller matrices. It's worth noting that understanding the concept of inverses helps us solve matrix equations similar to how reciprocals help solve algebraic equations in numerical systems. In the context of our problem, we used the inverse of \( \mathbf{A} \) to solve the equation \( \mathbf{A} \mathbf{B} = \mathbf{0} \), leading us to a logical deduction.

Zero Matrix

The zero matrix is akin to the number zero in regular arithmetic. It is a matrix in which every element is 0. Just as multiplying any number by zero results in zero, multiplying any matrix by a zero matrix results in a zero matrix of compatible dimensions.

• Notation for a zero matrix often simply uses \( \mathbf{0} \).
• Size matters: an \( n \times n \) zero matrix is different from any other dimension zero matrix.

In the given problem, \( \mathbf{A} \mathbf{B} = \mathbf{0} \) proved that when a nonsingular matrix multiplies another matrix to yield a zero matrix, then the other matrix \( \mathbf{B} \) must be a zero matrix. It is a crucial tool in linear algebra to determine null spaces and simplify systems of equations.

Matrix Multiplication

Matrix multiplication is an operation where two matrices are combined to produce another matrix. It plays a foundational role in linear algebra due to its various applications. Unlike regular multiplication of numbers, matrix multiplication isn't commutative, meaning \( \mathbf{A} \mathbf{B} eq \mathbf{B} \mathbf{A} \) in general.

• The resulting matrix dimensions depend on the original matrices: for matrices \( \mathbf{A} \) of size \( m \times n \) and \( \mathbf{B} \) of size \( n \times p \), the result is \( m \times p \).
• During multiplication, the element at position \((i, j)\) in the result is calculated as the sum of the products of elements from row \(i\) of \( \mathbf{A} \) and column \(j\) of \( \mathbf{B} \).

In the context of the exercise, understanding matrix multiplication allows us to interpret the product \( \mathbf{A} \mathbf{B} = \mathbf{0} \) and apply properties of matrix inverses to deduce further truths, such as \( \mathbf{B} = \mathbf{0} \). This operation is central for operations involving transformations, systems of equations, and data science.