Enter your matrix dimensions (rows and columns), fill in the matrix elements, and provide a scalar value — the Matrix by Scalar Calculator multiplies every element by that scalar and displays the full result matrix. Works with matrices up to 4×4. Also try the Eigenvalue and Eigenvector Calculator.
Result Matrix (scalar × A)
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Matrix Dimensions
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Scalar Applied
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Sum of Result Elements
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Largest Element in Result
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Smallest Element in Result
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To multiply a matrix by a scalar, simply multiply every single element in the matrix by that scalar value. For example, if your scalar is 3 and your matrix contains the element 4, the resulting element is 3 × 4 = 12. The dimensions of the matrix stay exactly the same. See also our Condition Number κ(A) — Condition Number.
Scalar multiplication follows several key properties: it is distributive over matrix addition — k(A + B) = kA + kB; it is associative — (xy)A = x(yA); multiplying by 1 leaves the matrix unchanged — 1·A = A; and multiplying by 0 produces a zero matrix. These properties make scalar multiplication a fundamental operation in linear algebra.
Dividing a matrix by a number n is the same as multiplying it by 1/n. Simply enter your divisor as a fraction (e.g., 0.5 to divide by 2) in the scalar field. Every element of the matrix will be divided by that number.
If you multiply an n×n matrix A by a scalar k, the determinant of the result is kⁿ · det(A). For example, multiplying a 3×3 matrix by scalar k gives a determinant of k³ · det(A). This is because each row of the determinant picks up a factor of k. You might also find our use the Tensor Product Calculator useful.
When you multiply a matrix A by scalar k, the eigenvalues of the resulting matrix kA are simply k times each eigenvalue of A. The eigenvectors, however, remain unchanged. So if λ is an eigenvalue of A, then kλ is an eigenvalue of kA.
Any matrix multiplied by the scalar 0 results in a zero matrix — a matrix of the same dimensions where every element equals 0. This is consistent with the property that 0 × anything = 0.
Multiplying the identity matrix I by a scalar k produces a scalar matrix — a diagonal matrix where each diagonal element equals k and all off-diagonal elements remain 0. This matrix acts like k when multiplied with other conformable matrices.
No, scalar multiplication is commutative with respect to the scalar itself. k·A = A·k, meaning the scalar can be written on either side of the matrix. However, this only applies to scalar multiplication; regular matrix-by-matrix multiplication is generally not commutative.