The Product of a 1x2 Matrix and a 2x2 Matrix

Oct 13, 2025 by wbrum.org 10 views

Introduction

In linear algebra, matrix multiplication is a fundamental operation that allows us to combine two matrices to produce a new matrix. The product of two matrices is a matrix that is obtained by multiplying the rows of the first matrix with the columns of the second matrix. In this article, we will explore the product of two matrices, specifically the product of a 1x2 matrix and a 2x2 matrix.

Matrix Multiplication

Matrix multiplication is a binary operation that takes two matrices as input and produces a new matrix as output. The product of two matrices A and B is denoted by AB and is defined as follows:

If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix.
The element in the ith row and jth column of the product matrix AB is calculated as the dot product of the ith row of matrix A and the jth column of matrix B.

The Product of a 1x2 Matrix and a 2x2 Matrix

In this article, we will consider the product of a 1x2 matrix and a 2x2 matrix. The 1x2 matrix is given by:

\left[\begin{array}{ll} 4 & 2 \end{array}\right]

and the 2x2 matrix is given by:

\left[\begin{array}{cc} -2 & 5 \\ 7 & -1 \end{array}\right]

To calculate the product of these two matrices, we need to multiply the row of the 1x2 matrix with the columns of the 2x2 matrix.

Calculating the Product

Let's calculate the product of the two matrices:

The first element of the product matrix is calculated as the dot product of the row of the 1x2 matrix and the first column of the 2x2 matrix:

\left[\begin{array}{ll} 4 & 2 \end{array}\right] \times \left[\begin{array}{c} -2 \\ 7 \end{array}\right] = 4(-2) + 2(7) = -8 + 14 = 6

The second element of the product matrix is calculated as the dot product of the row of the 1x2 matrix and the second column of the 2x2 matrix:

\left[\begin{array}{ll} 4 & 2 \end{array}\right] \times \left[\begin{array}{c} 5 \\ -1 \end{array}\right] = 4(5) + 2(-1) = 20 - 2 = 18

Therefore, the product of the two matrices is:

\left[\begin{array}{l} 6 \\ 18 \end{array}\right]

Conclusion

In this article, we have explored the product of a 1x2 matrix and a 2x2 matrix. We have calculated the product of these two matrices and obtained the result:

\left[\begin{array}{l} 6 \\ 18 \end{array}\right]

This result is consistent with the definition of matrix multiplication, which states that the product of two matrices is a matrix that is obtained by multiplying the rows of the first matrix with the columns of the second matrix.

Answer

The correct answer is:

\left[\begin{array}{l} 6 \\ 18 \end{array}\right]

This is the product of the matrices:

\left[\begin{array}{ll} 4 & 2 \end{array}\right] \times \left[\begin{array}{cc} -2 & 5 \\ 7 & -1 \end{array}\right]

Discussion

Matrix multiplication is a fundamental operation in linear algebra that allows us to combine two matrices to produce a new matrix. The product of two matrices is a matrix that is obtained by multiplying the rows of the first matrix with the columns of the second matrix. In this article, we have explored the product of a 1x2 matrix and a 2x2 matrix and obtained the result:

\left[\begin{array}{l} 6 \\ 18 \end{array}\right]

This result is consistent with the definition of matrix multiplication and demonstrates the importance of matrix multiplication in linear algebra.

References

Linear Algebra and Its Applications by Gilbert Strang
Matrix Algebra by James E. Gentle
Introduction to Linear Algebra by Gilbert Strang
Q&A: Matrix Multiplication =============================

Introduction

Matrix multiplication is a fundamental operation in linear algebra that allows us to combine two matrices to produce a new matrix. In our previous article, we explored the product of a 1x2 matrix and a 2x2 matrix and obtained the result:

\left[\begin{array}{l} 6 \\ 18 \end{array}\right]

In this article, we will answer some frequently asked questions about matrix multiplication.

Q: What is the condition for matrix multiplication to be possible?

A: Matrix multiplication is possible if the number of columns in the first matrix is equal to the number of rows in the second matrix.

Q: How do I calculate the product of two matrices?

A: To calculate the product of two matrices, you need to multiply the rows of the first matrix with the columns of the second matrix. The element in the ith row and jth column of the product matrix is calculated as the dot product of the ith row of the first matrix and the jth column of the second matrix.

Q: What is the difference between matrix multiplication and scalar multiplication?

A: Matrix multiplication is the operation of combining two matrices to produce a new matrix, while scalar multiplication is the operation of multiplying a matrix by a scalar (a number).

Q: Can I multiply a matrix by a vector?

A: Yes, you can multiply a matrix by a vector. The result is a vector that is obtained by multiplying the rows of the matrix with the vector.

Q: What is the property of matrix multiplication that states that the product of two matrices is not necessarily commutative?

A: The property of matrix multiplication that states that the product of two matrices is not necessarily commutative is known as the non-commutativity of matrix multiplication. This means that the order of the matrices matters, and the product of two matrices is not necessarily the same as the product of the matrices in reverse order.

Q: Can I multiply a matrix by itself?

A: Yes, you can multiply a matrix by itself. The result is a matrix that is obtained by multiplying the rows of the matrix with the columns of the matrix.

Q: What is the property of matrix multiplication that states that the product of a matrix with its transpose is a symmetric matrix?

A: The property of matrix multiplication that states that the product of a matrix with its transpose is a symmetric matrix is known as the symmetry of matrix multiplication. This means that if you multiply a matrix by its transpose, the result is a symmetric matrix.

Q: Can I multiply a matrix by a matrix that has a different number of rows and columns?

A: No, you cannot multiply a matrix by a matrix that has a different number of rows and columns. The number of columns in the first matrix must be equal to the number of rows in the second matrix for matrix multiplication to be possible.

Conclusion

Matrix multiplication is a fundamental operation in linear algebra that allows us to combine two matrices to produce a new matrix. In this article, we have answered some frequently asked questions about matrix multiplication and explored the properties of matrix multiplication.

References

Linear Algebra and Its Applications by Gilbert Strang
Matrix Algebra by James E. Gentle
Introduction to Linear Algebra by Gilbert Strang

Frequently Asked Questions

What is the condition for matrix multiplication to be possible?
How do I calculate the product of two matrices?
What is the difference between matrix multiplication and scalar multiplication?
Can I multiply a matrix by a vector?
What is the property of matrix multiplication that states that the product of two matrices is not necessarily commutative?
Can I multiply a matrix by itself?
What is the property of matrix multiplication that states that the product of a matrix with its transpose is a symmetric matrix?
Can I multiply a matrix by a matrix that has a different number of rows and columns?