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Chapter 9: Problem 67

What are equal matrices?

Short Answer

Expert verified
Two matrices are considered equal if they have the same dimensions (same number of rows and columns) and if each corresponding pair of elements in the two matrices are equal.

Step by step solution

01

Understand the Definition of Equal Matrices

Two matrices are said to be equal if they are of the same size and their corresponding elements are equal. That means, first, they must have the same number of rows and columns. Second, for each entry in the matrix, the corresponding entry in the other matrix must be the same. For instance, if we have two matrices A and B, they can only be equal if the size of A equals the size of B, and every element a_{ij} in A equals the corresponding element b_{ij} in B.
02

Defining Size of Matrices

When we say that two matrices, A and B, must be the same size, it means they must have the same number of rows and the same number of columns. Let's denote the size of a matrix with 'm' for the number of rows and 'n' for the number of columns. If matrix A is an m*n matrix, then matrix B must also be an m*n matrix.
03

Identifying Corresponding Elements in Matrices

Once we have established that the matrices are of the same size, we must prove that all corresponding elements are equal. In other words, every entry a_{ij} in matrix A must be exactly equal to the entry b_{ij} in matrix B, where 'i' denotes the row number, and 'j' denotes the column number. If this condition is satisfied for all entries, then matrices A and B are indeed equal.

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Most popular questions from this chapter

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{rrr} {1} & {-1} & {4} \\ {4} & {-1} & {3} \\ {2} & {0} & {-2} \end{array}\right], \quad B=\left[\begin{array}{rrr} {1} & {1} & {0} \\ {1} & {2} & {4} \\ {1} & {-1} & {3} \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ X-B=A $$

Exercises \(72-74\) will help you prepare for the material covered in the next section. In each exercise, refer to the following system: $$ \left\\{\begin{aligned} 3 x-4 y+4 z &=7 \\ x-y-2 z &=2 \\ 2 x-3 y+6 z &=5 \end{aligned}\right. $$ Show that \((12 z+1,10 z-1, z)\) satisfies the system for \(z=1\)

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ A-X=4 B $$

If two matrices can be multiplied, describe how to determine the order of the product.
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